Chapter 1. Social conventions: Hume, Lewis and game theory

The tradition of understanding social coordination as a source of social order is historically rich. Aristotle grounded social conventions in human nature and the pursuit of eudaimonia, or flourishing. He viewed humans as “political animals” who naturally form communities to achieve collective well-being. Justice and virtue, central to his ethics, were seen as the basis for political order. Unlike later followers of the social contract theory, Aristotle saw social organization as intrinsic to human rationality rather than a deliberate agreement (aristotle1998?).

Hobbes reimagined social conventions as constructs invoked by humanity’s violent “state of nature.” He argued that self-preservation drives individuals to surrender freedoms to an absolute sovereign via a social contract resulting from explicit agreement (Hobbes, 2016). Conventions thus arise from fear and rational self-interest, not innate sociability.

According to Epstein (2018), a notion of convention was first explicitly used as an alternative to agreement by Pufendorf (1673), to refer to language and law. He synthesized Hobbesian ideas with theological natural law. While agreeing that humans are self-interested, he attributed the “law of sociality” to divine mandate, requiring peaceful coexistence despite innate corruption. For Pufendorf, natural law obligates humans to form civil societies, with God as the ultimate author of social conventions. This introduced a moral dimension absent in Hobbes’s instrumentalist framework, suggesting that conventions are not merely utilitarian but also morally justified. His point was that conventions do not need to be explicitly agreed to and might exist and work without their intentional design. This intuition has remained largely unchanged.

Hume’s theory of social conventions, articulated most prominently in A Treatise of Human Nature (Hume, 2003) and An Enquiry Concerning the Principles of Morals (1751), offers a groundbreaking empiricist account of how social norms and institutions emerge organically from human interaction rather than rational design or divine mandate. Hume’s analysis hinges on three core premises:

• the role of custom in shaping behavior
• the centrality of mutual benefit in solving coordination problems
• the artificiality of conventions

These are seen as products of collective habit rather than explicit agreement. The components form the scaffolding of his theory, which bridges psychology, ethics, and political philosophy.

Hume’s empiricist framework posits that human understanding arises from sensory impressions and ideas derived from them. This extends to social behavior: conventions emerge not from reason but from repeated experiences that cultivate habits. For instance, Hume’s iconic example of two individuals rowing a boat illustrates how synchronization arises through trial and error, not prior negotiation:

“Two men who pull at the oars of a boat, do it by an agreement or convention, tho’ they have never given promises to each other”(Hume, 2003).

However, Schliesser (2024) stipulates that this kind of coordination is not backed by “Humean conventions” as they, according to Hume himself¹, require “positive social externality”, whereas two burglars could effectively row away from a crime scene. We will not focus on this morally-driven notion of conventions.

Over time, repeating patterns solidify into conventions because they resolve practical problems (coordinating labor, establishing property rights) while minimizing friction. Custom, as Hume writes, “renders our experience useful to us” by creating stable expectations about others’ behavior, even in the absence of formal rules (Hume, 2003) . This emphasis on habit challenges rationalist theories like Hobbes’s by showing how conventions evolve unconsciously through iterative adjustments.

Hume highlights four key features of conventions:

• Mutual benefit: all parties gain from adhering to the convention (e.g., synchronized rowing ensures progress; standardized currency facilitates trade)
• Multiple potential solutions: different solutions could theoretically work (e.g., rowing fast or slow), but consistency matters more than the specific choice
• Unplanned agreement: conventions develop spontaneously through “a slow progression” of trial and error, not deliberate contract
• Reciprocity: adherence depends on the expectation that others will reciprocate, creating a self-reinforcing cycle of trust.

For Hume, conventions like property rights arise because humans recognize the “common interest” in stabilizing possessions to avoid conflict, even if their natural inclinations lean toward self-interest (Hume, 1998). This pragmatic focus distinguishes his theory from moralistic accounts, framing conventions as tools for managing inherent human partiality.

Hume classifies conventions as artificial virtues, social constructs developed to counteract humanity’s “limited generosity”. Unlike natural virtues like benevolence, which arise instinctively, conventions like justice or promise-keeping require cultivation. Their artificiality, however, does not make them arbitrary. Instead, they gain normative force through collective sentiment: individuals approve of conventions that promote social utility, and disapproval of violations strengthens adherence over time. This process explains how conventions acquire moral weight, transforming into norms that feel binding even when rational self-interest might suggest defiance. Experimental studies inspired by Hume’s (or rather Lewis’s (1969)) work confirm that conventions stabilize behavior even when incentives to defect arise, underscoring the interplay of habit and normativity (Guala & Mittone, 2010).

Hume’s theory diverges sharply from social contract models. While Hobbes root conventions in deliberate agreements to escape chaos or secure rights, Hume dismisses the notion of a primordial “state of nature” requiring such pacts. Instead, he argues that conventions emerge incrementally from lived experience, reflecting his broader skepticism toward rationalist abstractions. His framework also anticipates modern game theory, particularly David Lewis’s analysis of conventions as coordination equilibria (Lewis, 1969), though Hume places greater emphasis on psychology.

Crucially, Hume’s account bridges descriptive and normative domains. By showing how conventions evolve from practical needs to moral norms, he offers a naturalistic explanation for social order that avoids appeals to divine law or metaphysical necessity. This aligns with his rejection of causation as anything beyond observed regularity, reinforcing his view that human institutions are contingent products of custom rather than eternal truths.

After Hume, philosophers in the Scottish Enlightenment held that social order is an emergent product of individuals’ interactions, however, no such order has been specifically intended by individuals. As Ferguson (1980) writes, “nations stumble on establishments which are, indeed, the result of human action, but not the execution of any human design”. Afterwards, however, the study of conventions has quieten.

Lewis has revived and operationalized Hume’s insights into a theory of conventions using game theory and treating conventions as equilibria sustained by common knowledge and precedent. While Hume emphasizes historical contingency and gradual emergence, Lewis imposes stricter criteria of rationality and mutual expectations (Lewis, 1969). He sees conventions as solutions to coordination problems, a class of problem in game theory, a branch of mathematics dealing with strategic behavior, which require two or more agents to align their actions to produce a jointly optimal outcome. In the next section, we will briefly tour game theory and its main concepts before getting back to Lewis’s theory of conventions as game theory will be crucially important in the remainder of the thesis.

Game theory background

Game theory is a mathematical framework used to analyze situations of strategic interaction between rational decision-makers. Originally developed by John von Neumann and Oskar Morgenstern in their seminal work Theory of Games and Economic Behavior (morgenstern1944?), game theory has since evolved to encompass a wide range of applications in economics, biology, political science, and sociology (Gintis, 2009a; osborne2004?). It provides the tools to study how individuals or groups make choices when their outcomes depend not only on their own decisions but also on the decisions of others. The fundamental building blocks of game theory are games, players, strategies, payoffs, and equilibria (Zamir, Maschler, & Solan, 2013).

A strategic game in game theory is defined as a formal model G = (N, S, P) where:

• N represents the set of players involved in the game
• S = (S_1, S_2, \dots, S_n) denotes the strategy sets of each player, where S_i is the set of strategies available to player i
• P = (P_1, P_2, \dots, P_n) specifies the payoff functions, where P_i: S_1 \times S_2 \times \dots S_n \rightarrow \mathbb{R} gives the utility for player i given the chosen strategy profile (myerson1991?).

A strategy s_i \in S_i is a complete plan of action a player will follow in any situation they might face within the game. Payoffs represent the rewards or utilities that players receive based on the combination of strategies chosen by all involved.

One of the central concepts in game theory is equilibrium, where no player has an incentive to unilaterally change their strategy given the strategies of others. The most well-known equilibrium concept is the Nash equilibrium (NE), introduced by John Nash in the early 1950s (nash1950?). A strategy profile (s_1^*, s_2^*, \dots, s_n^*) forms a Nash equilibrium if for every player i, the following condition holds:

P_i(s_i^*, s_{-i}^*) \geq P_i(s_i, s_{-i}^*) \quad \forall s_i \in S_i.

Here,

• P_i represents the payoff function for player i
• s_i^* denotes the strategy chosen by player i at equilibrium
• s_{-i}^* represents the combination of strategies chosen by all other players except player i
• The inequality states that player i cannot increase their payoff by unilaterally changing their strategy from s_i^* to any other available strategy s_i.

Shortly after Nash’s work, Robert Aumann introduced the concept of correlated equilibrium (CE) in 1974 (Aumann, 1974). This generalization of Nash equilibrium allows players to coordinate their strategies through signals from a trusted mediator. Unlike Nash equilibrium, where players act independently, correlated equilibrium enables communication or correlation of strategies, capturing coordination through shared information. In a correlated equilibrium, a random signal suggests a strategy to each player, and players follow the recommendation if it is in their best interest to do so. Formally, a correlated equilibrium satisfies:

\sum_{s'_{-i}} q(s_i, s'_{-i}) \cdot [P_i(s_i, s'_{-i}) - P_i(s'_i, s'_{-i})] \geq 0 \quad \forall s_i, s'_i.

Here,

• q(s_i, s'_{-i}) represents the probability that the mediator recommends strategy s_i to player i and s'_{-i} to the other players
• P_i(s_i, s'_{-i}) is the payoff to player i when they play s_i and the others play s'_{-i}
• The inequality ensures that the expected payoff from following the recommendation is at least as great as from deviating.

As Roger Myerson has reportedly observed, “If there is intelligent life on other planets, in a majority of them, they would have discovered correlated equilibrium before Nash equilibrium” (Solan & Vohra, n.d.). Correlated equilibrium can be a more natural concept than Nash equilibrium, as its mathematical simplicity and reliance on cooperation make it easier to discover. He argued that humanity’s prioritization of Nash equilibrium may have been an accident of history rather than a reflection of its fundamental importance. In societies or civilizations where cooperative behavior is emphasized or external mediators are prevalent, correlated equilibrium could emerge as a more intuitive starting point for understanding strategic interactions.

In the realm of evolutionary biology, John Maynard Smith introduced the concept of evolutionarily stable strategy (ESS) in 1973 (maynard1973?). An ESS is a strategy s^* that is robust against invasion by mutant strategies and satisfies the following condition:

P(s^*, s^*) > P(s', s^*) \quad or \quad [P(s^*, s^*) = P(s', s^*) \quad and \quad P(s^*, s') > P(s', s')].

Here,

• P(s^*, s^*) is the payoff when both the incumbent and the invader use strategy s^*.
• P(s', s^*) is the payoff when the invader uses strategy s' while the incumbent sticks to s^*.

Beyond Nash, correlated, and ESS equilibria, game theory explores other equilibrium concepts, including subgame perfect equilibrium, trembling hand perfect equilibrium, and proper equilibrium, among others. These refinements address limitations of the NE, particularly in dynamic and extensive-form games. Notable equilibrium refinements include:

• Subgame Perfect Equilibrium: introduced by Selten (selten1965?), it ensures rational behavior at every stage of a dynamic game by requiring equilibrium strategies in every subgame. It refines NE by eliminating non-credible threats and is particularly relevant in sequential games.

• Trembling Hand Perfect Equilibrium: proposed also by Selten (selten1975?), it accounts for the possibility of small, unintended mistakes (or trembles) by requiring that strategies remain optimal even if there is a slight probability of error. This refinement helps to eliminate equilibria that are not robust to slight deviations.

• Proper Equilibrium: introduced by Myerson (myerson1978?), this concept strengthens trembling hand equilibrium by further penalizing less likely mistakes. It ensures that less probable errors are assigned proportionally smaller probabilities, reinforcing stability.

• Sequential Equilibrium: developed by Kreps and Wilson (kreps1982?), this refinement addresses the problem of imperfect information by combining strategies with beliefs about what has happened earlier in the game. It is particularly useful in signaling games and dynamic strategic interactions.

• Perfect Bayesian Equilibrium: extending the Bayesian framework, it requires that players update their beliefs consistently using Bayes’ rule and choose optimal strategies given their beliefs. It is widely applied in games with incomplete information (fudenberg1991?).

These equilibrium refinements aim to ensure stability and plausibility in strategic settings by accounting for dynamic aspects, imperfect information, and potential errors.

Coordination and cooperation problems are fundamental challenges in social philosophy since Hobbes (2016), and game theory has been an indispensable tool for tackling these problems due to its clarity and rigor.

• Coordination problems arise when individuals or groups need to choose between multiple possible equilibria, creating ambiguity about which solution will be selected. These problems are central to strategic interaction because they reflect situations where all parties would benefit from making compatible choices but may struggle to agree on a single option.

• Cooperation problems, on the other hand, highlight the conflict between individual rationality and collective benefit, where mutual cooperation yields a better outcome for all, but self-interest may lead to suboptimal results. Such challenges often require mechanisms to facilitate coordination or encourage cooperation, including social conventions or equilibrium selection techniques. Consequently, equilibrium concepts are fundamentally linked to coordination and cooperation problems because they model how rational agents arrive at stable solutions given others’ strategies.

Examples of coordination and cooperation problems include classic games like the Battle of the Sexes and the Prisoner’s Dilemma. In the former, a husband and a wife coordinate on choosing a leisure activity where everyone is satisfied with the choice, and in the latter, two prisoners independently either defect or cooperate with each other by uncovering her partner in crime to an officer. The payoff matrices of these games are shown below².

\begin{array}{|c|c|c|} \hline & Football & Ballet\\ \hline Football & 2,1 & 0,0 \\ \hline Ballet & 0,0 & 1,2 \\ \hline \end{array}

Battle of the Sexes

\begin{array}{|c|c|c|} \hline & Cooperate & Defect \\ \hline Cooperate & -1,-1 & -3,0 \\ \hline Defect & 0,-3 & -2,-2 \\ \hline \end{array}

Prisoner’s Dilemma

These matrices model real-world problems such as social dilemmas and negotiations. For instance, the Battle of the Sexes often represents situations where partners must choose between competing preferences, while the Prisoner’s Dilemma models the challenge of mutual cooperation versus self-interest in scenarios like arms races or public goods provision.

To illustrate the practical difference of equilibrium concepts in solving coordination problems, let us consider the Battle of the Sexes with pure Nash, mixed Nash and correlated equilibria.

In pure Nash, two pure strategy equilibria exist: both players attend either Ballet or Football. These equilibria ensure perfect coordination but are inherently unfair, as one player always prefers the chosen event over the other.

A mixed strategy Nash equilibrium also exists, where players randomize their choices independently, but it risks miscoordination. Let the Husband choose Ballet with probability p and Football with 1-p, and let the Wife choose Ballet with probability q and Football with 1-q. Using the indifference principle according to which a player randomizes her strategies in a way that the opponent is indifferent between their own available strategies, we calculate probabilities:

1. For the Husband to be indifferent, the Wife’s mixed strategy must make his expected payoff from Ballet equal to that from Football: 2q + 0(1-q) = 0q + 1(1-q) \implies 2q = 1 - q \implies q = \frac{1}{3}

2. For the Wife to be indifferent, the Husband’s mixed strategy must make her expected payoff from Ballet equal to that from Football: 1p + 0(1-p) = 0p + 2(1-p) \implies p = 2(1-p) \implies p = \frac{2}{3}

Thus, in the mixed strategy Nash equilibrium:

• The Husband chooses Ballet with probability p = \frac{2}{3} and Football with 1-p = \frac{1}{3}.
• The Wife chooses Ballet with probability q = \frac{1}{3} and Football with 1-q = \frac{2}{3}.

The expected payoffs for both players in this equilibrium are:

• Husband: 2q + 0(1-q) = 2\left(\frac{1}{3}\right) + 0 = \frac{2}{3},
• Wife: 1p + 0(1-p) = 1\left(\frac{2}{3}\right) + 0 = \frac{2}{3}.

This mixed strategy equilibrium represents a compromise balancing fairness and coordination through randomization, albeit less efficient than pure Nash equilibria due to inherent miscoordination risks.

In contrast, correlated equilibria utilize external signals to coordinate actions effectively. For instance, a public signal such as a coin flip can recommend both players attend Ballet with 50% probability and Football with 50% probability. This mechanism eliminates miscoordination entirely and ensures equal expected payoffs for both players (1.5 each). Correlated equilibria can achieve higher payoffs and fairness compared to both pure and mixed Nash equilibria by leveraging shared randomness or communication.

To demonstrate how external signal affects the payoff structure, we add a new strategy “Follow Signal (FS)”, where players choose based on a fair coin flip (Heads = Ballet, Tails = Football). The payoffs depend on actual coordination, not just expectations: we can calculate expected payoffs when one player uses FS and the other does not.

FS (H) vs. Ballet (W):

• Signal = Heads (50%): Both choose Ballet → (2, 1).
  
• Signal = Tails (50%): H chooses Prize Fight, W stays at Ballet → (0, 0).
  
• Expected payoff: 0.5 \times (2, 1) + 0.5 \times (0, 0) = (1, 0.5).

FS (H) vs. Football (W):

• Signal = Heads (50%): H chooses Ballet, W stays at Prize Fight → (0, 0).
  
• Signal = Tails (50%): Both choose Prize Fight → (1, 2).
  
• Expected payoff: 0.5 \times (0, 0) + 0.5 \times (1, 2) = (0.5, 1).

Thus, the augmented game matrix becomes:

\begin{array}{|c|c|c|c|} \hline & Ballet (W) & Football (W) & FS (W) \\ \hline Ballet (H) & (2, 1) & (0, 0) & (1, 0.5) \\ \hline Football (H) & (0, 0) & (1, 2) & (0.5, 1) \\ \hline FS (H) & (1, 0.5) & (0.5, 1) & (1.5, 1.5) \\ \hline \end{array}

The strategy profile of (FS, FS) represents a Nash equilibrium because neither player has an incentive to deviate. If the Man switches to Ballet, he would only receive 1, a decrease from his current payoff of 1.5 when the Woman remains at FS. Similarly, if the Woman switches to Prize Fight, she would receive only 1, a decrease from her current payoff of 1.5 when the Man stays at FS. Since no profitable deviation exists for either player, the strategy profile (1.5, 1.5) is stable. Thus, the CE strategy is as an NE strategy of an augmented game. The difference is that CE are computationally simpler to compute than NE and model real-world scenarios where external signals (e.g., traffic lights) guide decisions. In summary, CE expand the solution space of a game, offering improvements over Nash equilibria when players can leverage a coordination device.

Getting back to coordination problems, O’Connor (2019) distinguishes two classes of them:

• correlative problems (same choice to coordinate)
• complementary problems (different choices to coordinate)

In correlative coordination problems, agents need to converge on the same choice to coordinate successfully. For example, consider a driving game, where two players drive towards each other and each can choose the left or right side to drive on. If they both are on the same side and no one swerves, they might crash, and if each of them chooses a different side, they will stay safe. One important feature of this and other coordination problems is arbitrariness, meaning that it does not matter on what side both players would converge. Instead, what matters is that they either coordinate by choosing the same action, for example, swerving to the right. On the game matrix, it is represented as two non-unique equilibria. It means that either of them solves the coordination problem.

\begin{array}{|c|c|c|} \hline & Swerve \quad left & Swerve \quad right \\ \hline Swerve \quad left & 1,1 & -1,-1 \\ \hline Swerve \quad right & -1,-1 & 1,1 \\ \hline \end{array}

Complementary coordination problems, as opposed to correlative ones, require from agents different actions, or strategies, to coordinate successfully. As O’Connor (2019) points out, division of labor or resources is an example of this class of games. For instance, two roommates want to organize a party and invite guests. To proceed, they need to tidy up the house and order pizza delivery. If they both do the cleaning, there will be no food when the guests come, and if they both order pizza delivery, they will have plenty of food but be embarrassed by the mess at the house.

\begin{array}{|c|c|c|} \hline & Order pizza & Tidy room \\ \hline Order pizza & -1,-1 & 1,1 \\ \hline Tidy room & 1,1 & -1,-1 \\ \hline \end{array}

The only difference between the two classes of coordination problems is either choosing same or different actions to coordinate successfully.

Coordination problems and conventions are intrinsically linked as former ones emerge when individuals or groups require aligned action for mutual benefit, necessitating communication and shared understanding to stabilize interactions. Conventions function as a mechanism for predictable coordination by encapsulating mutual expectations, thereby reducing ambiguity and establishing stable behavioral patterns within a social context. David Lewis’s theory of conventions as coordination equilibria, explored in the subsequent section, provides a central treatment of this relationship.

Intellectual influences of Lewis’s “Convention”

The intellectual atmosphere in which Lewis’s Convention was developed was mostly engaged with questions of language, meaning, and social behavior. Several intellectual movements and concerns shaped the development of his theory.

In the mid-20th century, the interest in influence of social practice on linguistic meaning kept growing, as philosophers like Quine (quine1960?) and Wittgenstein [-0@wittgenstein] argued that meaning arises from shared use within a community. Wittgenstein highlighted that language’s meaning emerges through public usage, rather than inherent semantic properties. For instance, “game” has no fixed definition but derives its meaning from the activities associated with it. Building on this tradition, Lewis sought to explain how linguistic conventions form, stabilize, and persist in communities by providing a systematic account of their development over time. By conceptualizing meaning as coordinated behavior, Lewis laid a foundation for viewing language as a socially orchestrated activity rather than an innate or purely individualistic construct. Consequently, communication relies not on objective meanings but on mutual expectations about usage, emphasizing convention’s crucial role in language (lewis1969?).

The Zeitgeist of analytic philosophy in the 1960s grappled with the legacy of Logical Positivism, which, through formal logic and empirical verification, defined meaning based on analytically true statements or verifiable empirical claims (Godfrey-Smith, 2003). However, by the 1960s, critiques from Quine, Putnam, and others challenged this framework, particularly the distinction of analytic/synthetic truths, the former being true in virtue of their meaning and the latter in virtue of their relationship to the world.

Quine rejected traditional notions of necessity and analyticity, asserting ontological commitments are embedded within theories and language (quine1951?; quine1960?; quine1969?), emphasizing empirical evidence and pragmatic considerations in shaping beliefs. His critique of analyticity underscored the revisability of language, highlighting conventions as mutable rather than fixed. Putnam’s “Twin Earth” thought experiment³ further developed these ideas, advocating semantic externalism—the view that word meaning depends on external facts, not solely on mental states—challenging internalist accounts of meaning and emphasizing the role of external factors in linguistic practices. Consequently, conventions are understood as influenced by contextual and environmental factors, moving beyond purely internal or necessary determinations.

Lewis’s theory of convention was a way to address this intellectual shift by emphasizing the contingent nature of meaning. Rather than being dictated by any necessity, conventions arise as arbitrary but stable solutions to coordination problems, reflecting a more pragmatic and flexible understanding of linguistic meaning and social practices. It highlights that even the most strict traditions started as flexible behavioral patterns which might have been otherwise but have been amplified more and more with each iteration. This perspective is deeply rooted in Quinean ideas about language as being subject to revision, adaptation, and negotiation within a community or culture.

Another major philosophical concern that Lewis addressed was the ontology of social rules and norms, profoundly influenced by Hume’s work. Lewis developed Hume’s idea of conventions emerging and persisting even in the absence of centralized enforcement. Lewis argued that conventions are self-reinforcing: once established, individuals have no reason to deviate as long as others continue to conform. The major deviation from Hume’s thought was accent on rationality of agents as the source of such conformity, whereas Hume emphasized psychological custom.

An example of this can be seen in the development of money as a medium of exchange. Initially, various objects—such as cattle, shells, or metal coins—served as currency. Over time, paper money became widely accepted, not because of any intrinsic value, but because people expected others to accept it in transactions. This insight was later influential in discussions of spontaneous order and decentralized systems in political philosophy and economics, particularly in the work of Hayek (hayek1973?). By explaining conventions as natural outcomes of repeated social interactions, Lewis contributed to a broader understanding of how norms, institutions, and linguistic practices can arise organically without explicit design or coercion.

Furthermore, Hume’s skepticism about moral realism, a position stating that objective moral norms exist, played a role in shaping Lewis’s view of conventions as arbitrary yet stable⁴. Hume argued that moral distinctions are not grounded in objective properties but in human sentiment and social conditioning. Similarly, Lewis contends that conventions are not determined by any intrinsic necessity but arise contingently through social practices. For instance, the choice of driving on the right or left side of the road is arbitrary, yet once established, it becomes self-reinforcing because all individuals benefit from adherence to the norm. This reflects Hume’s broader thesis that social order emerges not from absolute principles but from shared expectations and learned behaviors.

If the problems of meaning, language and conventionality served as the issue Lewis wanted to attack and Hume’s notion of convention was resource to build upon, Lewis still needed a tool to construct his argument with. He found it in game theory (vonneumann1944?) and, in particular, in Schelling’s approach to strategic interaction in “mixed motive” games. (Schelling, 1980).

Game theory offered a structured mathematical framework for analyzing strategic interactions among individuals conceived as rational actors. Lewis’s engagement with game theory and decision theory was facilitated by this prevailing intellectual trend. The emphasis on formal models and rational choice provided a common language and conceptual framework for discussing social behavior across diverse disciplines, making it a natural progression for a philosopher like Lewis to explore these powerful analytical tools in his own work.

Schelling’s work represented a significant departure from prevailing game theory’s emphasis on zero-sum conflict (when there is always a winner and a loser), recognizing that real-world interactions frequently exhibit “mixed motives” or simultaneous conflicting and converging interests. He critiqued the limitations of purely mathematical analysis of strategic interaction and advocated for empirical research to illuminate the conditions shaping behavior, specifically considering opportunities for communication and the presence of attractive alternatives. This expanded scope featuring both conflict and cooperation included the very phenomena of cooperation and coordination that drew Lewis’s attention in the context of the problem of social conventions.

Schelling argued that conflict and cooperation are not necessarily opposing forces but are deeply intertwined in strategic interactions. One of his key contributions was the concept of credible commitment, where the ability to commit to a particular strategy in advance can influence an opponent’s decisions (schelling1960?). A fundamental aspect of this is self-binding, where a player deliberately restricts her own options to strengthen bargaining position.

Another crucial insight was the concept of focal points (also known as Schelling points), which are solutions that individuals naturally gravitate toward in coordination games without explicit communication. Schelling demonstrated this through experiments where participants, when asked to choose a meeting place in New York City without coordination, overwhelmingly selected noon at Grand Central Terminal, although it was a location with no inherent payoff advantage but high cultural prominence (schelling1960?).

In the study of pure coordination games, Schelling examined interactions where players share interests but lack communication, such as selecting matching integers for a reward. Participants often converged on salient choices, such as the number 1, due to its distinctiveness as the smallest positive integer (schelling1960?). His work also refined the Nash equilibrium by demonstrating how focal points can help identify stable and salient outcomes among multiple NE (lewis1969?). Furthermore, for conflict scenarios, he introduced the concept of “threats that leave something to chance”, showing that probabilistic threats, such as partial mobilization, can deter adversaries more effectively than deterministic ones by leveraging uncertainty to maintain deterrence (schelling1960?).

Lewis formalized Schelling’s insights into a theory of conventions, defining them as solutions to recurrent coordination problems where agents align on focal points due to mutual expectations (lewis1969?). Conventions rely on extrinsic incentives, such as avoiding coordination failure, rather than intrinsic obligations. Lewis also emphasized that communication itself is a coordination game, where signals, such as Paul Revere’s lanterns, derive Meaning from shared conventions (lewis1969?).

One of the central ideas Lewis took from Schelling is the concept of focal point, or salience. He showed that social conventions arise as focal points for coordination. For instance, in many societies, people drive on one designated side of the road not because of an inherent preference for that side, but because universal adherence to a single convention ensures safety and predictability. Building on that idea, Lewis argues that agents select the most salient convention which “stands out” from alternatives, either through precedent, explicit agreement, or intrinsic properties. According to Lewis, salience, a subjective psychological trait independent of the strategic situation, governs convention emergence and conformity. Specifically, Lewis addresses how conventions arise (dynamics – through initial selection and subsequent salience amplification) and why people conform (statics – due to the overwhelming salience of a pre-existing convention, fostering an expectation of adherence). Subsequent refinements of Lewis’s theory reimagine and formalize the notion of salience mostly through evolutionary lens Skyrms (2014).

Another crucial concept Lewis adopts from Schelling is the role of expectation and self-enforcement in strategic equilibrium. Schelling showed that in many coordination scenarios, once an equilibrium is established, deviation becomes irrational since the costs of uncoordinated action outweigh potential individual gains. Lewis builds on this by defining conventions as self-perpetuating: once a convention is in place, individuals follow it not because of external enforcement, but because mutual expectations make deviation costly. This is evident in linguistic conventions, where the use of certain words and grammatical structures persists because everyone expects others to conform to them.

Furthermore, Lewis’s notion of common knowledge, foundational to his theory of conventions, derives from Schelling’s emphasis on mutual awareness within strategic interaction which is tightly connected with salience. Though Schelling lacked formalization, he highlighted the crucial role of shared understanding for successful coordination. Lewis expanded upon this, asserting that convention stability necessitates not just adherence, but also recognition as the expected behavior within a group, thereby enabling convention maintenance across generations and large populations.

By drawing on Schelling’s work, Lewis was able to provide a game-theoretic foundation for the study of conventions, demonstrating how they emerge, stabilize, and persist over time. Whereas Schelling’s focus was on strategic choices in conflict and negotiation, Lewis extended these principles to the domain of language, social norms, and epistemic coordination, thus broadening the applicability of game-theoretic insights to philosophy and social science. As a result, Schelling’s The Strategy of Conflict remains one of the key intellectual influences behind Lewis’s Convention and its enduring impact on theories of social coordination.

Lewis’s theory of conventions

Lewis’s analysis focuses on coordination problems—strategic situations where agents share a common interest in a mutually acceptable outcome, which necessitates matching choices predicated on expectations of others. Such problems involve multiple equilibria achievable through identical or different actions (as in correlative and complementary game classes of O’Connor (2019)), which demand agents to act solely based on anticipated behavior of others and are characterized by interdependent decision-making between at least two agents with a prevailing coincidence of interest.

Conventions, according to Lewis, solve coordination problems via salient and mutually-beneficial strategic choice options.

Lewis defines social conventions as an arbitrary yet self-sustaining behavioral pattern emerging from repeated coordination problems between two or more players. Its distinctive feature is players’ conformity to these behavioral patterns, for they expect others to do so, and it is common knowledge that every player is expected to conform. Deviation from a conventional choice of action leads to lower payoff, so players do not have incentives to deviate unilaterally which is on their own. For example, if everyone drives on the right side of the road, it is rational for each driver to do the same to avoid collisions. Lewis (lewis1969?) formulates convention as follows:

A behavioral regularity R within a population P in a repeated situation S qualifies as a convention if and only if:

1. Every member of P conforms to R.
2. Each individual expects others to conform to R.
3. All members have similar preferences regarding possible behavioral patterns.
4. Each person prefers universal conformity to R, provided that nearly everyone else adheres to it.
5. Members would also prefer an alternative regularity R' under the same conditions, as long as R' and R are mutually exclusive.

Lewis later refines his analysis to accommodate occasional deviations from convention and Skyrms (2023) even introduces quasi-conventions as unstable conventions based on yet another equilibrium concept of coarse correlated equilibrium. Despite this, much of the academic discourse focuses on the strict version of his definition.

Lewisian convention is a particular instance of NE (Nash equilibrium). In NE, no participant can improve their outcome by unilaterally changing their strategy. If deviation strictly reduces payoff, the equilibrium is considered strict. In this sense, NE represents a “steady state,” where each individual acts optimally given the actions of others. However, Lewisian convention extends beyond NE by emphasizing collective preference for conformity, even when minor deviations occur. Such equilibria are referred to as coordination equilibria.

Lewis’s framework highlights arbitrariness in conventions, where R is defined as a convention only if an alternative R' could serve equally well. This acknowledges that conventions are contingent choices among possible solutions rather than inherent necessities which continues the insights of Quine (quine1969?), Putnam (putnam1975?) and others.

Additionally, Lewis introduced the concept of common knowledge and made it a condition for a regularity to be a convention, where a fact p is common knowledge if:

• Everyone knows p.
• Everyone knows that everyone knows p.
• Everyone knows that everyone knows that everyone knows p, and so on.

This recursive understanding of knowledge has spurred extensive discussion in both philosophical and game-theoretic literature. Aumann (aumann1976?) and Schiffer (schiffer1972?) have developed formalizations of common knowledge, diverging from Lewis’s original informal approach.

As we will tour this and other aspects of Lewis’s theory in detail later in this chapter, it suffices to mention that further reception of his theory saw the common knowledge requirement too cognitively demanding and unrealistic (Bicchieri, 2005; Binmore, 2008; Gilbert, 1992; P. Vanderschraaf, 1998; camerer2003?).

As Lewis’s theory uses game theory, rationality plays a fundamental role in Lewis’s framework. He assumes that agents are instrumentally rational, meaning they choose actions that maximize their expected utility given their beliefs and expectations about the world and the behavior of others. Although the entire metaphor of humans as maximizing agents has been questioned (Paternotte, 2020), it still serves as guidance in economic theory (Gintis, 2007b; Gintis & Helbing, 2013), biology (Engel & Singer, 2008; S. Okasha, 2017; Samir Okasha & Binmore, 2012) and human ecology (Mouden, Burton-Chellew, Gardner, & West, 2012; Sterelny, 2012). However, there are alternative views on the requirement of agent’s rationality for conventions to exist. Ruth G. Millikan (2022) suggests that conventions stabilize only by the weight of precedent, thus not requiring any rationality or consciousness. We will look closer at such alternative in the third chapter of the thesis while discussing complex signaling as a source of the transition from ‘animal conventions’ to human social institutions.

Lewis’s notion of conventions weaves behavior, beliefs, preferences, and expectations into a framework of common knowledge and rationality to explain the stability of conventions. Each part of the definition is vital: common knowledge ensures a shared understanding of the convention, the preference for conformity incentivizes adherence given others’ cooperation, and rationality guides individual choices within the context of shared expectations.

As a primary motivation for Lewis’s analysis was to address the philosophical problem of linguistic meaning, he aimed to argue that language is grounded in conventions which do not require up-front agreement on terms. Just as drivers coordinate on a side to drive without a formal contract, speakers of a language develop conventions of using sounds or gestures to refer to specific things through repeated interaction and mutual expectations. Lewis viewed language as a system of signaling, where meaning arises from the conventional association between signals (words, phrases) and states of the world. For example, the word “cat” conventionally signals the presence of a feline. This convention is sustained because speakers generally intend to be truthful and listeners generally trust that they are being told the truth. This mutual expectation and reliance on the regularity of signal-meaning pairings allows for effective communication, which is a form of coordination.

This led Lewis to delineate behavioral and signaling conventions (Lewis, 1969, pp. 147–150), where the former coordinate actions and the former coordinate meaning. As a prototypical example of a signaling convention, Lewis gives a story of Paul Revere and the lanterns hung in the steeple of the Old North Church used to warn colonial militia about approaching British Troops in 1775. Two hung lanterns conveyed that troops are advancing by sea, one—by land. Additionally, the actions of the receiver of the message, given each of these signals, would differ. In other words, senders and receivers of a message coordinate on following a pre-established pattern of “if X, do Y” like in the example with lanterns⁵.

For Lewis, signaling conventions are a special case, or a subclass, of behavioral conventions as they share basic properties like arbitrariness, conformity and being common knowledge. Signaling conventions differ in that they involve communication and interpretation of meaning and solve coordination problems by information transfer. They require encoding/decoding which is producing and interpreting signals.

An important feature of the relationship between these two classes of conventions is that, according to Lewis, signaling conventions fundamentally rely upon and are shaped by pre-existing behavioral conventions. For example, language meanings of words depend on both parties’ adherence to established norms of pronunciation and grammar. Signaling systems frequently exhibit nesting, where specific conventions are embedded within larger behavioral regularities. For instance, raising one’s hand to speak during a meeting is a signaling conventions nested within a broader behavioral convention of turn-taking.

There is a formal distinction between behavioral, or “general” as Lewis call it, and signaling conventions. In signaling games, the players can be either senders or receivers, where the former owns private information about the world state and send a signal about it and the latter observes the signal and acts on it. More formally, it looks like the following:

1. World states: L (left) and R (right)
2. Signals: V₁ and V₂
3. Actions: Aᴸ (left action) and Aᴿ (right action)

\begin{array}{|c|c|c|} \hline & R₁ & R₂ \\ \hline S₁ & (1,1) & (0,0) \\ \hline S₂ & (0,0) & (1,1) \\ \hline \end{array}

If a sender’s signal representing a world state is correctly acted upon by receiver, both parties get the payoff of (1, 1) and if either party fails to map (“encode” or “decode”) information, they get (0, 0). There is a plethora of possible options within this informational “layer” of signaling system extensively studied primarily by philosophers of biology (Godfrey-Smith, 1991; Huttegger & Skyrms, 2008; Shea, Godfrey-Smith, & Cao, 2018; Skyrms, 2010a, 2010b).

Godfrey-Smith (2014) refined Lewis’s model by distinguishing state-act and act-act coordination, where in the former signals map states to receiver action and in the latter they synchronize action between agents without any external events. Act-act coordination allows to view Hume’s boat rowers as an act-act signaling system: the rowers’ rhythmic strokes serve as imperative signals (“Row now!”) that directly coordinate mutual actions rather than conveying information about external conditions (Martínez & Godfrey-Smith, 2016). The absence of an exogenous state reduces the system to a pure coordination game employing Nash or coordination equilibrium, where the “signal” (stroke rhythm) functions as a self-reinforcing convention stabilized by common interest and reciprocal expectations. Unlike state-dependent signaling of state-act coordination, which requires alignment between acts and external facts, act-act systems like the rowboat prioritize interpersonal synchronization through real-time behavioral feedback, illustrating how communication can organize joint action without representational content.

A paradigmatic real-world example of a state-act signaling system is alarm calls specific for each type of predator. For example, vervet monkeys have a call for for seeing eagles which conveys hiding in the grass and a call for seeing a snake conveying climbing on a tree (Seyfarth & Cheney, 1990). A perfect connection between a world state, signal and action comprises a signaling system.

Although formally similar, as both behavioral and signaling conventions can be described as games with players and payoffs, they differ in that the latter have an additional “layer” of information between players. And although Lewis himself proclaimed signaling conventions a subcategory of behavioral ones, the relationship between them is not clear. For Skyrms, signals inform action, and signaling networks coordinate action, which implicitly conveys signaling conventions as underpinning behavioral ones. Skyrms further suggests that signaling is responsible for the evolution of teamwork itself (Skyrms, 2010b), which questions Lewis’s hierarchical categorization and creates a version of a chicken-and-egg problem. We will look closer at the relationship between behavioral and signaling conventions and its role in emergence of social institutions in the third chapter.

Criticisms and problems Lewis’s theory generated

Lewis’s theory has been criticized on many grounds, and, as Rescorla (2024) notes, virtually every component of his theory has been under attack: from imprecise notion of equilibrium concept to the very necessity of conventions for solving coordination problems. Many criticisms have been met in refinements and extensions of Lewis’s theory by later scholars.

There are five main areas of criticism of Lewis’s account of conventions:

1. conformity requirement and hidden normativity
2. overestimation of arbitrariness
3. common knowledge requirement and source of salience
4. connection between conventions and coordination problems
5. imprecise equilibrium concept

We will survey 1-4 here as 5 is an extension rather than critique which we will address in the next section on refinements of Lewis’s theory. Each subsection starts with immediate criticism of Lewis’s theory and continues with a larger problem related to Lewis’s theory this criticism points to.

Hidden normativity of conventions

One of the major criticisms of Lewis’s theory of conventions is unrealistic conformity requirement expressed of his 4-th clause: “each person prefers universal conformity to R, provided that nearly everyone else adheres to it”. As some scholars points out, this strict requirement rules out such regularities as sending thank-you notes after dinner (Gilbert, 1992) as non-conventional, for they do not require complete conformity. Many commentators find this unintuitive as we usually call any mutually expected behavioral regularity a convention regardless of its level of conformity.

However, a possible defense of Lewis’s position is to restrict a social group where convention takes place and to add that “each person within a certain social group prefers universal conformity to R…”. This addition addresses Gilbert’s criticism in that it supports an idea of near-complete conformity relative to the scale and size of a social group with operative convention. If sending thank-you notes after a dinner within a certain group is indeed a convention, not writing such a note would at least disappoint a dinner host. Of course, this might not impose any external sanctions on a guest not writing a thank-you note. However, conformity relative to group size highlights inherent normativity in the form close to normative expectation, which Bicchieri (2005) considers an essential ingredient of social norms rather than conventions.

As can be seen, a convention’s level of conformity poses a deeper problem, that of normativity of conventions. The level of conformity helps distinguish between conventions as regularities de facto and de jure (Rescorla, 2024), where the former describe actual behavior and the latter prescribe how individuals should behave in certain situations. Lewis himself anticipated such objections and claimed that conventions eventually become social norms. This claim later generated a major controversy over the relationship between those (Bicchieri, Muldoon, & Sontuoso, 2023). Level of conformity points to a problem of the source of such conformity, which concerns Lewis’s critics.

Gilbert (1992) contends that Lewis’s account ignores the normative force of conventions. For Gilbert, the source of conformity of conventions is joint commitments that bind participants to collective ends, creating obligations to conform and justifying criticism of defectors. Lewis’s model, which reduces conventions to equilibrium strategies in coordination games, cannot explain why individuals feel obliged to comply with conventions (e.g., stopping at red lights) or apologize for breaching them. Gilbert also disputes Lewis’s focus on coordination problems, arguing that many conventions like etiquette rules lack clear coordination benefits and instead reflect shared commitments. In Lewis, there is appeal to instrumental rationality which maximizes expected value and avoids sanctions, but some scholars see this as insufficient to substantiate conventions.

For instance, Guala & Mittone (2010) study the extent to which Lewis conventions are normative. He addresses both theoretical and empirical aspects and conclude that Lewis has put forward a scientific theory of conventions and not an analysis of the folk notion, and that conventions do indeed have intrinsic normativity beyond that of instrumental rationality. However, there is another strand of scholars that disagree and put forward that the only normativity of conventions is that of instrumental rationality (Bacharach & Bernasconi, 1997; Gold & Sugden, 2007).

In experimental settings, given an iterated Ultimatum or Prisoner dilemma game, only 29% of potential deviants in the lab choose to breach emergent convention (Guala & Mittone, 2010). As Guala and Mittone argue, players in these experiments may unintentionally create extra pressure to conform with their shared history of action, beyond the requirements of rational decision-making and social norms. However, the exact mechanism of additional normative expectation formation is to be discovered.

From this, Guala concludes that Lewis’ model provides an incomplete account of conventions’ ontology. Data suggest that Lewis Conventions acquire normative force through repeated play, and any future model must account for this. This has non-trivial implications for theory and practice, as it implies that habits and customs may be hard to disrupt.

In a similar vein, Hindriks (2019) claims that instrumental rationality cannot motivate adherence to conventions and norms and their perception as legitimate. Instrumental rationality with its costs and expected utilities fails to capture the motivation by the normative part of convention itself and not by the costs of its violation. Hindriks claims that it is normative expectations and normative beliefs that complement sanctions as a source for norm existence and perception as legitimate.

Overestimation of arbitrariness

Overestimation of arbitrariness is another area of criticism. According to Lewis, arbitrariness is one of the key distinguishing aspects of conventions. However, as Gilbert (1992) points out, not all possible solutions to a coordination problem are equally profitable for players. In cases where one way of coordinating is more preferred than another, convention will not be that arbitrary. In other words, alternative conventions are logically justified, but pragmatically implausible as there is almost always a slight “preference” of one convention over the other due to different factors like historical accident and history of play. Later scholars talked about this in terms of symmetry-breaking by stochastic events (Skyrms, 2010b, 2010a) and salience of conventions amplified by the history of iteratively playing a certain coordination game (Korbak, Zubek, Kuciński, Miłoś, & Rączaszek-Leonardi, 2021).

Arbitrariness was recast as a continuum between contingency and necessity, or conventionality and functionality (O’Connor, 2019). Signaling between vervet monkeys might well be modeled as a convention in the Lewisian sense of repeated behavioral patterns of solving coordination problems (Harms, 2004; Skyrms, 2010b). However, this convention is not historically contingent in the sense of several possible solutions being equally profitable as Lewis supposes and as Gilbert critiques, for there are evolutionary constraints breaking the symmetry between multiple equilibria. Agents might be (and most probably are) hardwired to following certain strategies in certain environmental conditions. This distinction, as O’Connor underlines, highlights some conventions as more functional and others as more arbitrary.

A similar line of criticism comes from Burge (1975), who notes that Lewisian requirement for convention to involve mutual knowledge of alternative regularities, or practices that could replace existing ones if widely adopted, is too strict. Conventions might fix without agents’ knowledge of alternatives, Burge argues. He contends that conventions can stabilize with habit, custom or tradition, widely following Hume’s original argument, and that knowledge of alternative conventions is not needed. Conventions, as Burge argues, are not governed by any biological, psychological or sociological law, they are historically accident. In addition, agents do not necessarily deliberate to “switch” from one convention to another. In terms of game theory, Lewis requires that agents know the structure of the game with its multiple equilibria, whereas Burge’s notion does not. This leads to yet another point of criticism, overly intellectualist requirements for agents.

Epistemic overreach of common knowledge requirement

Common knowledge denotes an epistemic state within a group wherein a proposition p is known by all members, and each member knows that every other member knows p, recursively extending to an infinite level of iterated knowledge. This recursive nature differentiates it from mere mutual knowledge, which necessitates only that each individual knows p. Consequently, common knowledge represents an idealized, stringent condition profoundly impacting coordination and strategic interaction, prompting investigation into its feasibility and real-world relevance.

As Cubitt & Sugden (2003) underline, Lewis’s initial conception of common knowledge did not imply unconstrained cognitive capacity of idealized agents. As they put forward, a proposition p is common knowledge if a state of affairs A exists where everyone has a reason to believe A holds, A indicates to everyone that everyone has a reason to believe A holds, and A indicates to everyone that p. This definition generates an infinite chain of “reasons to believe” rather than an infinite chain of “knowledge,” suggesting a more pragmatic approach towards achieving coordination. This approach acknowledges the limitations of human epistemic capabilities and focuses on the justification for beliefs about states of affairs and others’ beliefs about them rather than in absolute certainty on every level of iterated knowledge. Nevertheless, the majority of scholars interpret Lewisian conventions as computationally and cognitively demanding.

Gilbert (1992) criticized the infinite regress of Lewis’s common knowledge. She challenged the psychologically implausible requirement of infinite levels of iterated knowledge, arguing it is unnecessary for explaining social phenomena like collective belief and convention. Gilbert proposed a framework centered on joint commitment, asserting that social facts emerge from situations where individuals are collectively committed to intend or believe something as a unified body, rather than through an infinite chain of individual beliefs about others’ beliefs. This joint commitment involves a shared intention or belief held by a group as a collective entity, irrespective of individual members’ personal convictions—for instance, a group’s shared commitment despite private doubts. This approach provides a means to understand shared social states and collective actions, generating shared obligations and expectations that drive behavior and shape attitudes, thereby avoiding the demanding epistemic requirements of common knowledge.

(bicchieri1993?) argued that real-world agents operate under bounded rationality, which is more psychologically plausible. Individuals possess finite processing capacity and memory, which makes an infinite regress of knowledge untenable. Bicchieri investigated how agents form beliefs and expectations about others’ actions in coordination games, emphasizing mutual expectations and the potential for coordination through learning and repeated interactions, even without full common knowledge. She highlighted the role of social norms, proposing that they function through conditional preferences – individuals preferring to conform if they expect others to do so – and normative expectations, which are beliefs about what others believe one ought to do. This allows coordination to emerge and persist through observation, belief updating, and conformity, irrespective of the norm’s common knowledge status.

(heifetz1999?) underscored the limitations of the common knowledge assumption in dynamic settings and games with temporal imprecision where communication is not instantaneous or unreliable. The coordinated attack problem when two parties agree to attack at the same time exemplifies how the absence of guaranteed, instantaneous communication can preclude the establishment of common knowledge, leading to suboptimal outcomes. Researchers have investigated alternative, weaker notions like finite levels of mutual knowledge or common belief to account for imperfections in real-world information and bounded rationality, offering potentially more accurate models of coordination and cooperation.

One of the more radical criticisms of the common knowledge requirement comes from evolutionary game theory, a branch of game theory pioneered by Maynard Smith (1982) which assumes natural selection and evolutionary dynamics as a source of solutions for strategic games instead of rationality of self-interested actors with complete information. These criticisms doubt the necessity of common knowledge for conventions.

For example, Binmore (2008) challenged the infinite levels of common knowledge posited by Lewis, arguing that agents only require first-order expectations regarding others’ behavior to converge on an equilibrium. This perspective emphasizes accurate prediction of actions as a critical element for coordination, with rational players responding accordingly. Binmore’s evolutionary approach highlighted cultural evolution’s role in shaping these common understandings and norms, suggesting societies develop and transmit effective coordination strategies over time based on promoting social stability – a dynamic process which refines coordination strategies rather than a static, pre-existing condition of full common knowledge. He also notes that Lewis’s analysis of conventions confines its usage to small-scale societies as it implies observing public events being observed by another party. And this is not realistic in larger populations. Binmore suggests that conventions do not generally require common knowledge overall and can be established in evolutionary environments with only one level of reasoning instead of infinite hierarchy of beliefs. He also notes that everyday conventions mostly operate via automatic behavior and low-level mutual expectations.

Guala (2020) put forward a similar argument about “belief-less” coordination where most everyday conventions do not require iterated beliefs and hence cognitive capacities for meta-representation. Means-ends rationality and cheap heuristics are said to be sufficient.

Connection between conventions and coordination problems

Some scholars argue that conventions are not necessary for solving coordination problems, undermining Lewis’s theory. Sugden (2005) and P. Vanderschraaf (1998) argue that conventions need not necessarily be solutions for coordination problems—fashion or property conventions are not like this, for example⁶. Both of them have developed generalized accounts which do not require conventions to solve coordination problems. (davis2003?), Marmor (1996, 2009), (miller2001?), Sugden (sugden1986?; sugden2004?) have argued that conventions need not be coordination equilibria.

Sugden (2005) posits that conventions arise from behavioral patterns generating mutual advantage, independent of explicit coordination, thus rejecting Lewis’s focus on purely coordinating problems. Drawing on Hayek and Hume, he emphasizes spontaneous order of conventions which challenges the primacy of “constitutive” pre-establihed rules like law in governing social interactions. He argues that conventions emerge when patterns of behavior yield benefits for all participants, even in competitive or asymmetric situations. Unlike traditional game-theoretic models that focus on Nash equilibria, Sugden’s framework accommodates scenarios where no clear equilibrium exists which renders Lewis-style coordination problems too restrictive.

Sugden introduces the concept of team reasoning, where individuals act on collective goals rather than individual incentives, akin to Gilbert’s joint commitment and collective intentionality but without endorsing a “plural subject” ontology. Fashion conventions emerge through independent adoption of trends perceived as advantageous for social signaling. This framework elucidates conventions in competitive scenarios lacking coordination equilibria, exemplified by property rights systems governed by historical precedent rather than coordinated agreement.

As (davis2003?) and Marmor (2009) note, people follow trends for social distinction rather than coordination, yet these patterns become conventional through repeated adoption.

Marmor (2009) challenges Lewis’s emphasis on coordination problems, arguing instead for an analysis grounded in actual games like chess, distinct from the theoretical “games” favored by game theorists. His main argument is that there are deeper conventions like truth-telling which make Lewis-style coordination possible. He outlines three conditions for a rule to be considered conventional:

1. A population P normally follows rule R in circumstances C.
2. There is a reason A for members of P to follow R in circumstances C.
3. There exists at least one alternative rule S, such that if members of P had followed S instead of R, A would still have been a sufficient reason for following S, partly because S was generally followed instead of R. Rules R and S are mutually exclusive in the given circumstances.

Marmor draws two distinctions:

• coordination / constitutive conventions
• “deep” / “surface” conventions

Coordination conventions solve Lewis-style problems like driving sides by aligning actions for mutual benefit, depend on shared expectations and mutual compliance. Constitutive conventions create social practices or classes thereof like chess rules which constitute the game of chess itself/ Marmor argues that constitutive conventions emerge as responses to complex social needs and are foundational to many practices, including legal systems. Unlike coordination conventions, they do not depend on mutual expectations but instead define the ontology of the practice. Deep conventions are foundational norms that underpin social practices and are less amenable to change. For example, truth-telling is a deep convention necessary for effective communication. In its turn, surface conventions are more specific instantiations of deep conventions and vary across contexts. For instance, particular linguistic rules like grammar are surface conventions based on deeper norms like truth-telling.

Ruth Garrett Millikan (2005) presents a radically biological perspective on conventions, diverging significantly from economic and sociological approaches. Her core argument posits that a convention is fundamentally a behavior pattern sustained within a population through the mechanism of replicated precedent. Notably, Millikan rejects the prevailing tradition, exemplified by Hume and Lewis, which attributes social order to the rational decisions of individual agents. She explicitly denies any role for rationality in convention maintenance, asserting that a society upholding a convention solely through unreflective conformity would fulfill her definition. While Burgé similarly emphasizes factors beyond enlightened self-interest—including inertia, superstition, and ignorance—Millikan’s position is more extreme, entirely excluding any rational underpinning for convention stability.

For Millikan, conventions persist through replication adjusted according to the weight of precedent, where current patterns derive from prior instances. They are arbitrary and contingent as their stability is dictated neither by optimal design nor conformity of the majority. Instead, it influenced by its effective functional performance which might have been achieved with other patterns and does not require conscious adherence to rules. Millikan’s approach characterizes conventions as descriptive regularities which are emergent, stabilized patterns replicated through unconscious imitation, allowing for flexible adaptation without rigid definitions or universal agreement. For example, language speakers do not consciously follow a rule when calling a book a “book”,they simply replicate the behavior they have observed and linguistic conventions can be disobeyed without incurring sanctions, unlike rules in a normative sense. This contrasts with Lewis’s high-demanding view of mutual expectations, common knowledge and inherent normativity.

Millikan distinguished three types of coordination:

• blind coordination, where participants act without knowledge of each other’s actions (e.g., traffic systems in Lewisian examples)
• half-blind coordination, where one party anticipates the other’s behavior based on precedent (e.g., linguistic communication)
• open coordination, where both parties fully anticipate each other’s actions.

Linguistic conventions predominantly fall into half-blind coordination.

Millikan’s biological perspective frames conventions as analogous to evolutionary processes:

• Just as genes propagate based on their fitness, cultural conventions proliferate because they serve useful functions for individuals or groups.
• The “proper function” of a convention is its capacity to achieve specific outcomes (e.g., facilitating communication) effectively over time.

The notion of function will be important later as it is used in contemporary theories of social institutions as strategic equilibria (Guala & Hindriks, 2015) which try to smuggle biological functions and generate major controversy over the very notion and its relation to convention.

Extensions and refinements

Lewis’s theory of conventions became a starting point for formal research on conventions and later scholars refined his theory, sometimes to an unrecognizable extent. There are many refinements, but we will consider only most important for the topic of emergence of social institutions from animal conventions. In this section, we survey theories explicitly citing Lewis as a baseline.

As I mentioned in the previous section, imprecise equilibrium concept was among the popular criticisms of Lewis’s theory, and this component has been actively worked and elaborated on. Two notable reformulations of conventions are as correlated equilibria (CE) and evolutionary stable strategies (ESS).

Vanderschraaf’s inductive deliberation as a source of salience

Vanderschraaf (1995, 2001; 1998) redefined social conventions as CE through inductive learning, positioning conventions as foundational to achieving justice as mutual advantage. He formalized the notion of salience (or focal points) as information partitions and employed the Dirichlet rule⁷ to show how agents sequentially update their beliefs about others’ strategies to gradually arrive at an equilibrium.

Lewis considered a coordination equilibrium a convention if the players have common knowledge of mutual expectations. Vanderschraaf calls this mutual expectation criterion (MEC). Each agent has a decisive reason to conform to her part of the convention, expecting the other agents to do likewise. Lewis stated that an equilibrium must be a coordination equilibrium to reflect the notion that a person conforming to a convention wants their intention to be seen as such. Vanderschraaf calls it the public intentions criterion (PIC). Furthermore, Lewis argues that common knowledge of the MEC is necessary for a convention. However, as Vanderschraaf notes, it is not sufficient, since common knowledge of the MEC can be satisfied at any strict Nash equilibrium.

According to Vanderschraaf, a convention constitutes a strategy profile \sigma^* = (\sigma_1^*, \ldots, \sigma_n^*) where each agent i maximizes expected utility such that \mathbb{E}[u_i(\sigma_i^*, \sigma_{-i}^*)] \geq \mathbb{E}[u_i(\sigma_i', \sigma_{-i}^*)] for all alternative strategies \sigma_i' \neq \sigma_i^*, ensuring stability against unilateral deviations.

The formation of conventions operates not through cognitively expensive rational deliberation, but through relatively cheap inductive learning mechanisms. Agents employ Dirichlet dynamics to update beliefs about opponents’ strategies. This updating process describes how agents repeatedly revise their beliefs by incorporating new observations of others’ behavior. A deliberational equilibrium is then defined as a fixed point of this learning dynamic, where agents’ beliefs stabilize. The stabilized joint beliefs and strategies that emerge from this iterative updating correspond to what Vanderschraaf calls endogenous correlated equilibrium (ECE)⁸: a CE arising internally from the agents’ inductive learning and mutual belief revision, rather than from an external correlation device as it is usually presented in broader game theory literature⁹. Kôno (2008) has mathematically proven how ECE is possible and that distributions of ECE and exogenous CE are completely different. The Dirichlet dynamics responsible for arriving at ECE is modeled as follows:

p_{t+1}(s_{-i}) = \frac{n_{s_{-i}} + \alpha_{s_{-i}}}{\sum_{s'_{-i}} (n_{s'_{-i}} + \alpha_{s'_{-i}})}

where n_{s_{-i}} represents observed strategy profiles and \alpha_{s_{-i}} denotes prior beliefs (Peter Vanderschraaf, 2018). Repeated interactions lead to path-dependent emergence of focal points, particularly in bargaining scenarios. Two prominent conventions arise: equal division of goods (x_i = \frac{1}{n}) and egalitarian payoff distributions satisfying u_i(x_i) - u_i(d) = u_j(x_j) - u_j(d) for all agents i,j, where d represents disagreement payoffs (Peter Vanderschraaf, 1995).

An important part of Vanderschraaf’s theory of conventions is his contribution to moral philosophy and theory of justice. He grounded principles of justice in conventions that generate Pareto improvements¹⁰ over non-cooperative baselines. A just convention \sigma^J must satisfy u_i(\sigma^J) \geq u_i(\sigma^B) for all agents i, where \sigma^B denotes the baseline equilibrium (Peter Vanderschraaf, 2018).

This requirement addresses the vulnerability objection to justice theories which fail to adequately protect the most vulnerable persons. It does so by ensuring that conventions benefit even the least advantaged participants, creating mutual advantages that stabilize social arrangements. The framework reconciles Humean conventionalism with game theory, demonstrating how justice emerges from repeated coordination problems rather than abstract moral principles.

As can be seen, convention as CE allows for the “fair” coordination, even though no pure strategy equilibrium exists as we saw earlier with the “Battle of Sexes” game example. To reiterate, neither of the pure strategy Nash equilibria in this game is “fair”, in the sense that the players receive the same payoff.

This game has a mixed Nash equilibrium at which Husband plays A1 with probability \frac 2 3 and Wife plays A2 with probability \frac 2 3, and at this equilibrium each player’s expected payoff is \frac 2 3, so this equilibrium is “fair”. However, at the mixed Nash equilibrium, both players are indifferent to the strategies they play given what each player believes about her opponent, so this equilibrium fails the PIC and is consequently not a convention. Nevertheless, there is a correlated equilibrium fair to both players, and which each player will prefer over the pure strategy equilibrium that is unfair to her.

This game has a mixed Nash equilibrium at which both agents play their strategies with probability \frac 2 3, yielding an expected payoff of \frac 2 3 for each agent. However, this equilibrium does not satisfy the PIC and is thus not a convention. Nevertheless, there is a correlated equilibrium that is fair to both players and preferable to the pure strategy equilibrium. With a toss of a fair coin, there is a probability space \Omega = \{H, W\} with “heads” and “tails”. The agents have a common information partition \mathscr{H} = \{\{H\},\{W\}\} and the correlated strategy combination is denoted as a function f: \Omega \rightarrow \{A 1, A 2\} \times \{A 1, A 2\} with f(H) = (A 1, A 1) and f(W) = (A 2, A 2). Husband has a higher expected payoff with this combination than any of the other strategies, so she will not deviate from it. The expected payoff for Husband is 2 if the outcome is H, and 1 if it is W.

\begin{aligned} & \left.E\left(u_1 \circ f \mid H\right)=2>0=E\left(u_1(A 2, A 1)\right) \mid H\right), \text { and } \\ & E\left(u_1 \circ f \mid W\right)=1>0=E\left(u_1(A 1, A 2) \mid W\right) \end{aligned}

The same holds for the second player. To this end, neither player would want to deviate, since the overall expected payoff at this equilibrium for each player is

E\left(u_k \circ f\right)=\frac{1}{2} \cdot E\left(u_k \circ f \mid H\right)+\frac{1}{2} \cdot E\left(u_k \circ f \mid T\right)=\frac{3}{2}

It means that each player prefers the expected payoff from f to that of the mixed equilibrium.

For Vanderschraaf, a convention as a mapping of “states of the world” to strategy combinations of a noncooperative game (Peter Vanderschraaf, 1995, p. 69):

DEFINITION 1. A game \Gamma is an ordered triple (N, S, \mathbf{u}) consisting of the following elements:

1. A finite set N ={\{1,2, …, n\}}, called the set of players;

2. For each player k \in N, there is a finite set S_{k}= \{{A_{k_{1}}, A_{k_{2}},\dots, A_{kn_{k}}}\}, called the alternative pure strategies for player k. The Cartesian product S = S_{1} \times \dots \times S_n is called the pure strategy set for the game \Gamma;

3. A map \mathbf{u}: S \rightarrow \mathbb{R}^n, called the payoff function on the pure strategy set. At each strategy combination \mathbf{A} = (A_{1j_1}, \dots, A_{nj_{n})}\in S, player k’s payoff is given by the kth component of the value of \mathbf{u}, that is, player k’s payoff u_k, at \mathbf{A} is determined by u_k(\mathbf{A}) = I_{k} \circ \mathbf{u} (A_{1j_1}, \dots, A_{nj_n}),

where I_k(\mathbf{x}) projects \mathbf{x} \in \mathbb{R}^n onto its kth component.

As Vanderschraaf builds on Aumann’s model (1987), each player has a personal information partition \mathscr{H}_k of a probability space \Omega. Elementary events on \Omega are called states of the world. At each state \omega, each player k knows which element H_{kj}\in \mathscr{H}_k has occurred, but not which \omega. H_kj represents k’s private information about the states of the world. While k knows the opponent partitions, she does not know their content. A function f: \Omega \rightarrow S defines a exogenously correlated strategy n-tuple, such that at each state of the world \omega \in \Omega, each player k selects a strategy combination f(\omega)=(f_1(\omega),\dots,f_n(\omega))\in S correlated with the state of the world \omega. Thus, by playing f_k(\omega), k follows Bayesian rationality and maximizes expected payoff given private information and expectations regarding opponents.

DEFINITION 2. Given \Gamma = (N, S, \mathbf{u}), \Omega, and the information partitions \mathscr{H} of \Omega as defined above, f:\Omega \rightarrow S is a correlated equilibrium if and only if, for each k \in N,

1. f_k is an \mathscr{H}_k-measurable function, that is, for each H_{kj}\in \mathscr{H}_k, f_k(\omega) is constant for each \omega' \in H_kj, and

2. For each \omega \in \Omega, E(u_{k} \circ f|\mathscr{H}_k)(\omega) \geq E(u_{k} \circ (f_{-k}, g_k)|\mathscr{H}_k)(\omega)

where E denotes expectation, ‘-k’ refer to the result of excluding the kth component from an n-tuple. This holds for any \mathscr{H}_k-measurable function g_{k}: \Omega \rightarrow S_k. The correlated equilibrium f is strict if and only if the inequalities are all strict.

The measurability restriction on f_k means that k knows her strategy in each \omega. This definition implies that players have common knowledge of the payoff structure, partitions of \Omega, and f: \Omega \rightarrow S, which is needed to compute expected payoffs and reach correlated equilibrium. In addition, if the players possess common knowledge of Bayesian rationality, they will follow their ends of f, expecting others to do the same, since they jointly maximize expected utility in this way.

The agents refer to a common information partition of the states of the world. While each agent k has a private information partition \mathscr{H}_{k} of \Omega, there is a partition of \Omega, namely the intersection \mathscr{H}=\cap_{k \in N}\mathscr{H}k, of the states of the world such that for each \omega \in \Omega, all the agents will know which cell H(\omega) \in \mathscr{H} occurs. The agents’ expected utilities in the following Definition 3 are conditional on their common partition \mathscr{H}, reflecting the intuition that conventions rely upon information that is public to all.

The agents’ expected utilities are conditioned on their common information common partition \mathscr{H} of the states of the world, which is the intersection of all their private partitions \mathscr{H} = \cap_{k \in N}\mathscr{H}_k. This reflects that conventions depend on information available to all agents.

DEFINITION 3. Given \Gamma=(N, S, \mathbf{u}), \Omega, and the partition \mathscr{H} of \Omega of events that are common knowledge among the players, a function f: \Omega \rightarrow S is a convention if and only if for each \omega \in \Omega, and for each k \in N, f_k is \mathscr{H}-measurable and

E\left(u_k \circ f \mid \mathscr{H}\right)(\omega)>E\left(u_k \circ\left(f_{-j}, g_j\right) \mid \mathscr{H}\right)(\omega)

for each j \in N and for any \mathscr{H}-measurable function g_j: \Omega \rightarrow S_j.

It means that if any player j deviates from a convention f, every player k \in N, including j, will be worse off. This definition of convention as a strict correlated equilibrium satisfies the PIC, as all agents are aware of the common partition and the strategies each player is expected to play. Thus, if any opponent mistakenly thinks that a player k will play a strategy g_k(\omega) \neq f_k(\omega) other than the one prescribed by f, they may be tempted to deviate, resulting in a worse-off outcome for k. Conversely, if all opponents are aware that k will play her strategy f_k(\omega) at each state of the world \omega \in \Omega, then they have a strong incentive to conform with convention f(\omega), which gives k an improved outcome.

Overall, Vanderschraaf’s contribution is formalization of salience, hence he uses the common information partition \mathscr{H} as a necessary restriction to make the definition of convention conform with Lewis’ spirit. The other question is how salience itself emerges. Lewis suggests that pre-game communication, precedent, and environmental cues may lead agents to link their expectations and actions with various “states of the world”, thus achieving correlated equilibrium. However, these sources of salience face the problem of infinite regress, for it is unclear how precedent or pre-game communication occurred in the first place without an established and shared conventional rules. Vanderschraaf, along with Skyrms (Peter Vanderschraaf & Skyrms, 1993), proposes inductive deliberation as a mechanism by which salience is being established. It requires agents to be Bayesian rational and works by recursive belief modification. Players can reach a correlated equilibrium without communication by dynamically updating their beliefs using a common inductive rule, even if their beliefs don’t initially allow for an equilibrium.

Another significant extension of Lewis’s theory is related to redefining conventions as ESS and is due mostly to Skyrms.

Skyrms’s evolutionary approach to conventions

Skyrms integrated Lewis’s theory of conventions into an evolutionary framework. He showed how signaling conventions can emerge naturally with adaptive processes like evolution and learning in agents with limited cognitive sophistication which overcomes Lewis’s reliance on common knowledge (Skyrms, 2010b).

Although Skyrms has almost established an entire fruitful research program with many followers (Franke & Wagner, 2014; Huttegger, 2007a, 2007b; LaCroix, 2020a; O’Connor, 2020) and we will take a closer look at his generalization of Lewis’s signaling models later in this section, I suggest he would not have done it without his earlier and less-known contribution to game theory which has to do with generalization of the ESS solution concept.

The ESS, or evolutionary stable strategy, being a foundational solution concept in evolutionary game theory formulated by Smith & Price (1973) is a strategy that, if adopted by majority of population, cannot be invaded by any mutant strategy. Crucially, this concept implies random matching¹¹, where individuals are paired for strategic interactions independently of their types, such that the probability of encountering any strategy is only proportional to its overall population frequency. While this assumption simplifies analysis and yields elegant theoretical results, it limits the applicability of ESS to well-mixed populations and fails to capture the complexity of structured or socially embedded interactions.

Skyrms recognized that ESS does not generate stable strategies with non-random matching arising from mechanisms like kin selection, signaling systems, spatial or social structure. These correlations induce interactional dependencies increasing the probability of similar-strategy encounters. Such dependencies drastically alter the evolutionary dynamics and can stabilize strategies such as cooperation or signaling conventions that would be unstable or unsustainable under classical ESS assumptions (Skyrms, 1994).

This led Skyrms to establishing “adaptive ratifiable strategy” as a generalization ESS that incorporates the endogenous structure of interactions, making it a more realistic predictor of evolutionary outcomes. A strategy is adaptive-ratifiable if it maximizes expected fitness when it is nearly fixed in the population, taking into account the conditional probabilities of interacting with other strategies. This concept ensures dynamic stability under replicator dynamics¹² where correlation affects interaction frequencies (Skyrms, 1994).

Skyrms’s approach to conventions differs from Lewis’s in not relying on common knowledge and substituting it with evolutionary pressures which make conventions arise and persist. He showed that even simplest organisms like bacteria can arrive at signaling systems akin to Lewisian conventions with aid of simple adaptive mechanisms like mutation-selection or reinforcement learning (Skyrms, 2014).

Skyrms explored various learning dynamics that enable signaling systems to emerge in populations. For example:

• Simple Reinforcement Learning where agents adjust their strategies based on trial-and-error feedback from successful interactions. In a basic Lewis-Skyrms signaling game setup with 2 world states, 2 signals and 2 actions, senders and receivers begin with random dispositions and gradually reinforce successful pairings between states, signals, and actions.

• Win-Stay/Lose-Shift Dynamics where agents establish conventions more rapidly than simple reinforcement learning. This dynamic involves sticking with successful strategies while shifting away from unsuccessful ones, enhancing convergence speed and stability.

Skyrms’s framework models conventions as stable equilibria of sender-receiver games that evolve via reinforcement learning and evolutionary dynamics rather than rational deliberation. Formally, a signaling game involves:

• a set of states of the world S = \{s_1, s_2, \ldots, s_n\}
• a set of signals M = \{m_1, m_2, \ldots, m_k\}
• a set of acts A = \{a_1, a_2, \ldots, a_l\}.

The sender observes a state s \in Sand chooses a signal m\in Mto send. The receiver, upon receiving m, chooses an action a \in A. The payoffs $u_S(s, m, a) $and u_R(s, m, a) for sender and receiver respectively depend on how well the receiver’s action matches the state. Unlike Lewis’ model, which assumes common knowledge of salience to coordinate on a unique equilibrium, Skyrms shows that conventions can emerge through adaptive processes even when initial behaviors are random and no focal points exist.

A central concept in Skyrms’ analysis is the informational content of signals, which he quantifies using information-theoretic measures. Given a prior probability distribution over states P(S_i) and a posterior distribution conditioned on a signal $ m $, denoted $ P(S_i m) $, the information conveyed by $ m $ can be expressed as the vector of log-likelihood ratios:

\left( \log_2 \frac{P(S_1 \mid m)}{P(S_1)}, \log_2 \frac{P(S_2 \mid m)}{P(S_2)}, \ldots, \log_2 \frac{P(S_n \mid m)}{P(S_n)} \right).

where P(S_i) represents prior probabilities of states and P(S_i \mid m) denotes posterior probabilities conditioned on a signal m. This formalization bridges Lewis’s conceptual framework with mathematical models of communication.

This measure captures how a signal updates the receiver’s conditional strategy choice given the state of the world, thereby guiding action selection (Skyrms, 2010b).

Skyrms further explores signaling equilibria under conditions of partial alignment or conflict of interests between sender and receiver. In such cases, the equilibrium strategies may involve deceptive or partially informative signals. Formally, if the sender’s payoff function u_S differs from the receiver’s u_R, the equilibrium concept extends to signaling equilibria where strategies \sigma_S: S \to \Delta(M) and \sigma_R: M \to \Delta(A) satisfy mutual best responses:

\sigma_S(s) \in \arg\max_{m \in M} \mathbb{E}_{a \sim \sigma_R(m)}[u_S(s, m, a)], \quad \sigma_R(m) \in \arg\max_{a \in A} \mathbb{E}_{s \sim P(\cdot \mid m)}[u_R(s, m, a)],

where \Delta(X) denotes the set of probability distributions over X (skyrms1996?).

The evolutionary dynamics driving the emergence of conventions are often modeled through reinforcement learning algorithms such as the Roth-Erev model (Erev & Roth, 1998). Agents maintain propensities q_{i}(x) for choosing actions x (signals or responses), which are updated iteratively according to received payoffs:

q_{i}^{t+1}(x) = q_{i}^t(x) + \alpha \cdot \left( r_i^t(x) - q_i^t(x) \right),

where \alpha is a learning rate and r_i^t(x) is the reward at time t for action x (Skyrms, 2010b). Over repeated interactions, these learning dynamics lead to convergence on stable signaling conventions without requiring explicit coordination or rational foresight.

Transmission of information in signals and emphasis on informational content of a signal beget an lively response from philosophers of biology critiquing Skyrms for the lack of causation (Godfrey-Smith, 2020; Harms, 2004; Shea et al., 2018) which we will survey in Chapter 3.

An interesting part of Skyrms’s extension of Lewis signaling game is it’s implicit reliance on epistemic language of “observing” states of of the world and “interpreting” signals for “updating beliefs”. Although Skyrms utterly rejects any Bayesian interpretation of his signaling games (LaCroix, 2020b), he is sometimes interpreted as a incurring epistemology to his agents, especially when his theory is discussed side-by-side with natural theories of mental content (Baraghith, 2019; Harms, 2004; Ruth Garrett Millikan, 1987; Ruth Garrett Millikan & Millikan, 2004): that senders “represent” world states and transmit this public representation to a receiver who then “interprets” it with its own mental states. Consider vervet monkeys’ alarm calls. They can easily be described as involving mental states of “representing” an eagle and sending a certain signal to fellows monkeys who “decode” that public representation and map it onto suitable action. While plausible and the case for most natural theories of mental content like Ruth Garrett Millikan & Millikan (2004), it is not the case for Skyrms.

Although the structure of Lewis-Skyrms game mirrors the flow of information in epistemic contexts (state-signal-action pairings) and it is tempting to treat senders and receivers as Bayesian-rational, the Skyrmsian agents update their behavioral dispositions rather than beliefs as they do not possess any inference and can only adjust their mappings according to failure rates (Skyrms, 2012).

Skyrms’s sender-receiver system is an information channel focusing on how effective codes (signal-meaning pairings) arise and stabilize, not on agents’ beliefs or intention. His signaling games are mechanistic as Maynard Smith’s, for they take into account only objective, or “ontic”, features of agents like strategy frequency across population or, in case of signaling game, mappings from state to signal and from signal to action in accordance to the rate of coordination failures. Compare Lewis-Skyrms game

\begin{array}{ccccc} World & \xrightarrow{state} & Sender & \xrightarrow{Message} & Receiver & \xrightarrow{act} & {} \\ \end{array} \\

with Shannon’s information channel:

\begin{array}{ccccc} Source & \xrightarrow{original \quad message} & Encoder &\xrightarrow{signal} & Channel & \xrightarrow{signal} & Decoder & \xrightarrow{decoded \quad message} & {} \\ \end{array}

As Martínez (2019) proposes a “channel-first” view on signaling games and argues, the central behavioral unit of Lewis-Skyrms games is not strategies, but the encoding-decoding pair which is similar to mappings from above.

ОСТАНОВИЛСЯ ЗДЕСЬ

• вставить описание и критику скрирмса из Martínez (2019)

1. Correlation is a fundamental factor that alters evolutionary game dynamics

Skyrms argues that the standard assumption of random pairing in evolutionary game theory is often unrealistic because biological and social interactions tend to be correlated. This correlation changes the expected payoffs and thus the evolutionary trajectories of strategies, leading to outcomes that differ markedly from classical models. By incorporating correlation, evolutionary models better capture real-world phenomena such as kin selection and spatial structure.

“Where we have correlation, being an evolutionarily stable strategy in Maynard Smith’s sense is neither necessary nor sufficient for being a dynamically stable equilibrium.” (p. 2)

2. Adaptive-ratifiability is the appropriate stability criterion in correlated evolutionary games

Skyrms introduces adaptive-ratifiability as a refinement of the classical Evolutionarily Stable Strategy (ESS) concept. A strategy is adaptive-ratifiable if it maximizes expected fitness when it is nearly fixed in the population, taking into account the conditional probabilities of interacting with other strategies. This concept ensures dynamic stability under replicator dynamics where correlation affects interaction frequencies.
> “A pure strategy is ratifiable if it maximizes expected fitness when it is on the brink of fixation.” (p. 3)
Annotation: Adaptive-ratifiability generalizes ESS by incorporating the endogenous structure of interactions, making it a more realistic predictor of evolutionary outcomes.

3. Jeffrey’s decision theory framework is isomorphic to correlated evolutionary game theory

Skyrms adapts Richard Jeffrey’s logic of decision, which allows acts to influence the probabilities of states, to evolutionary contexts. This adaptation shows that expected fitness calculations in correlated games correspond to Jeffrey’s expected utility framework, establishing a deep conceptual link between rational decision-making and evolutionary processes.
> “Three characteristic features of Jeffrey’s discussion—expected utility, status quo utility, and ratifiability—play important parts in correlated evolutionary game theory.” (p. 4)
Annotation: This unification enables importing decision-theoretic insights into evolutionary biology, enriching both fields.

4. Correlation can lead to the selection of strictly dominated strategies

Contrary to classical evolutionary theory where dominated strategies are eliminated, Skyrms shows that under certain correlated interaction structures, strictly dominated strategies may persist or even be selected. This challenges the assumption that dominance relations alone determine evolutionary stability.
> “A strictly dominated strategy may be selected under conditions of correlation.” (p. 5)
Annotation: This result highlights the nuanced effects of interaction patterns, emphasizing that strategic dominance is context-dependent.

5. Perfect correlation enforces the selection of strictly efficient strategies

When interactions are perfectly correlated, populations evolve toward strictly efficient strategies that maximize collective payoffs. This outcome contrasts with uncorrelated settings where suboptimal equilibria often persist.
> “Under conditions of perfect correlation, a strictly efficient strategy must be selected.” (p. 6)
Annotation: Perfect correlation aligns individual incentives, effectively solving coordination problems by eliminating inefficient equilibria.

6. Correlation unifies diverse biological mechanisms promoting cooperation

Skyrms’s framework provides a unified theoretical treatment of various biological and social mechanisms—such as kin selection, population viscosity, signaling, and reciprocal altruism—that generate correlation in interactions and thereby facilitate cooperative behavior.
> “The resulting theory unifies the treatment of correlation due to kin, population viscosity, detection, signaling, reciprocal altruism, and behavior-dependent contexts.” (p. 7)
Annotation: This synthesis advances understanding of how different evolutionary forces converge to produce similar cooperative outcomes.

7. Classical ESS is insufficient to guarantee dynamic stability in correlated games

Skyrms demonstrates that Maynard Smith’s ESS criterion fails to ensure that a strategy will be an attractor under replicator dynamics when correlation is present. Adaptive-ratifiability better captures the notion of evolutionary stability in these more complex settings.
> “Being an evolutionarily stable strategy in Maynard Smith’s sense is neither necessary nor sufficient for being a dynamically stable equilibrium.” (p. 8)
Annotation: This insight calls for a re-evaluation of evolutionary stability concepts in light of realistic interaction structures.

8. Correlation enables cooperation in social dilemmas like the Prisoner’s Dilemma

In the classical Prisoner’s Dilemma with random pairing, defection dominates. Skyrms shows that introducing correlation—through mechanisms like signaling or kinship—can stabilize cooperative strategies by structuring interactions so cooperators preferentially meet cooperators.
> “Starting from any mixed population, the replicator dynamics with random pairing converges to a population of 100% defectors… but correlation can stabilize cooperative strategies.” (p. 9)
Annotation: This result provides a formal explanation for the evolution of cooperation in nature despite incentives to defect.

9. Incorporating mutation transforms deterministic replicator dynamics into stochastic processes

Skyrms notes that adding mutation to replicator models yields stochastic dynamics that better reflect biological realities. These stochastic models capture the probabilistic nature of strategy changes and the influence of correlation on evolutionary trajectories.
> “The desirable step of incorporating mutation into the model leads from deterministic dynamics to stochastic process models.” (p. 10)
Annotation: This extension allows modeling of evolutionary stability in fluctuating environments and finite populations.

Summary

Skyrms’s Darwin Meets the Logic of Decision fundamentally reshapes evolutionary game theory by incorporating correlation into interaction structures and linking evolutionary stability to decision-theoretic concepts. His introduction of adaptive-ratifiability replaces classical ESS in correlated settings, providing a more accurate criterion for dynamic stability. The paper unifies diverse biological mechanisms under a single theoretical umbrella and explains how correlation facilitates cooperation and efficient outcomes. This work bridges evolutionary biology and decision theory, enriching both disciplines with new conceptual tools.

% %\begin{array}{c|cc} % & A & B \\ \hline %A & (1,1) & (0,0) \\ \hline %B & (0,0) & (x,x) %\end{array} %

References