M Equilibrium: A dual theory of beliefs and choices in games

11/13/2018 ∙ by Jacob K. Goeree, et al. ∙ 0

We introduce a set-valued generalization of Nash equilibrium, called M equilibrium, which is based on ordinal monotonicity - players' choice probabilities are ranked the same as the expected payoffs based on their beliefs - and ordinal consistency - players' beliefs yield the same ranking of expected payoffs as their choices. Using results from semi-algebraic geometry, we prove there exist a finite number of M equilibria, each consisting of a finite number of connected components. Generically, M-equilibria can be "color coded" by their ranks in the sense that choices and beliefs belonging to the same M equilibrium have the same color. We show that colorable M equilibria are behaviorally stable, a concept that strengthens strategic stability. Furthermore, set-valued and parameter-free M equilibrium envelopes various parametric models based on fixed-points, including QRE as well as a new and computationally simpler class of models called μ Equilibrium. We report the results of several experiments designed to contrast M equilibrium predictions with those of existing behavioral game-theory models. A first experiment considers five variations of an asymmetric-matching pennies game that leave the predictions of Nash, various versions of QRE, and level-k unaltered. However, observed choice frequencies differ substantially and significantly across games as do players' beliefs. Moreover, beliefs and choices are heterogeneous and beliefs do not match choices in any of the games. These findings contradict existing behavioral game-theory models but accord well with the unique M equilibrium. Follow up experiments employ 3 by 3 games with a unique pure-strategy Nash equilibrium and multiple M equilibria. The belief and choice data exhibit coordination problems that could not be anticipated through the lens of existing behavioral game-theory models.

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1 Introduction

The central solution concept of game theory, the Nash equilibrium, rests on the assumption of mutually consistent behavior: each player’s choice is optimal given others’ choices. While the Nash equilibrium is defined in terms of a fixed-point condition on choices (Nash, 1950), it is quiet about how they come about. Presumably, players form beliefs about others’ behavior and use these to optimize their choices. Seen this way, the Nash assumption of mutually consistent behavior is equivalent to the following two conditions: players’ choices are best responses to their beliefs and players’ beliefs are correct.

Observed choices conform to Nash-equilibrium predictions in some settings, but a large body of work in experimental game theory has documented systematic departures from Nash equilibrium (e.g. Goeree and Holt, 2001). Nasar (1998) describes how the lack of experimental support for his equilibrium concept caused Nash to lose confidence in the relevance of game theory after which he turned to pure mathematics in his later research. Selten, who shared the 1994 Nobel Prize with Nash, likewise concluded that “game theory is for proving theorems, not for playing games.”

This paper is motivated by the desire for an empirically relevant game theory. We introduce a novel solution concept, equilibrium, which replaces the assumption of perfect maximization (“no mistakes”) with an ordinal monotonicity condition – players’ choice probabilities are ranked the same as their associated expected payoffs – and the assumption of perfect beliefs (“no surprises”) with an ordinal consistency condition – players’ beliefs yield the same ranking of expected payoffs as their choices. If the latter condition holds, which does not require beliefs to be homogeneous or correct, we say that players’ beliefs support their choices.

equilibrium is a set-valued solution concept that puts beliefs and choices on an equal footing. Specifically, equilibrium considers all beliefs that support a certain equilibrium choice and, dually, it considers all choices supported by a certain equilibrium belief. An equilibrium consists of the largest belief set and the largest choice set such that choices satisfy ordinal monotonicity and beliefs support choices.

We prove there exists an equilibrium for any normal-form game. We explore the geometry of equilibrium sets, which are examples of semi-algebraic sets (see e.g. Coste, 2002). Borrowing results from semi-algebraic geometry, we show there are finite number of

equilibria, each consisting of a finite number of connected components. Generically, there can be an even or odd number of

equilibria, unlike fixed-point theories, such as Nash equilibrium, that generically yield an odd number of equilibria (e.g. Harsanyi, 1973). We show that Nash equilibria arise as limit points of the -equilibrium sets. There may be fewer, as many, or more equilibria than Nash equilibria. Moreover, an equilibrium may contain zero, one, or more Nash equilibria. Importantly, we show the measure of the -equilibrium choice set falls quickly with the number of players and the number of possible choices.

Generically, any equilibrium can be “color coded” by the ranks of the equilibrium-choice profiles in the sense that all choices and beliefs that belong to the same equilibrium must have the same color. To illustrate, consider the symmetric two-player game in Table 1 where choice and belief profiles are compositions of red (), yellow (), and blue (). This “Mondrian game” has three symmetric equilibria: the colored “planes” in the left panel of Figure 1 show the -equilibrium choice sets, and the right panel shows the corresponding -equilibrium belief sets. Beliefs of a certain color support actions with the same color. The thick interior lines reflecting payoff indifferences form the boundaries of the colored planes.222Together with the three lines (not shown) that divide the simplex in six equal parts, i.e. the three lines where two of the three choice probabilities are equal. Conform Mondrian’s quote, drawing these lines is somewhat superfluous as all relevant information is encoded by the planes, including the Nash equilibria. For example, since all three colored planes include a vertex of the choice simplex, each color corresponds to a pure-strategy Nash equilibrium. Furthermore, the point on the edge of the choice simplex where the red (or yellow) plane borders the white plane corresponds to a degenerate mixed-strategy Nash equilibrium.333If denotes the symmetric equilibrium profile then the red boundary point is and the yellow boundary point is . These are the five symmetric Nash equilibria of the Mondrian game.

, , ,
, , ,
, , ,

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Table 1: Mondrian’s “composition of Red, Yellow, and Blue” game.
Figure 1: The symmetric -equilibrium choice sets (left) and belief sets (right) for the Mondrian game in Table 1.

We show that, generically, the interior of the equilibrium sets consists of choices and beliefs that are behaviorally stable. Roughly speaking, an -equilibrium profile is behaviorally stable when small errors in implementation or perception do not destroy its equilibrium nature. In other words, an -equilibrium profile is behaviorally stable when the profile is also an -equilibrium profile for all nearby games. Obviously, the concept of behavioral stability builds on ideas from the literature on refinements of Nash equilibrium (e.g. Van Damme, 1991), in particular, Kohlberg and Mertens’ (1986) “strategic stability.” The latter requires that nearby games have a Nash-equilibrium profile that is close to, but not necessarily equal to, the Nash-equilibrium profile of the original game. Behavioral stability strengthens this requirement by insisting that perturbations of the game do not change the set of -equilibrium profiles. We believe this stronger requirement is needed for equilibrium to be empirically relevant.

We establish equilibrium as a “meta theory” of various parametric models that rely on fixed-point conditions. In particular, we introduce a class of -Equilibrium models in which choice probabilities are parametric functions of the ranks of their associated expected payoffs. -Equilibrium choices follow from a fixed-point condition and -equilibrium beliefs support the fixed-point choice profile. We show that -equilibrium choices are easy to compute and typically supported by a continuous set of beliefs. Moreover, we prove that an -equilibrium of a given color contains all different -equilibrium fixed points supported by beliefs of the same color. Conversely, by varying , the different -equilibrium fixed-points “fill out” the -equilibrium choice set.

We report the results of a series of laboratory experiments that test equilibrium. The first experiment considers five variations of an asymmetric-matching pennies game that leave the predictions of various behavioral game-theory models (Nash, QRE, and level-) unaltered. However, observed choice frequencies differ substantially and significantly across games as do players’ beliefs. Moreover, beliefs do not match choices, and beliefs and choices are heterogeneous in any of the games. These findings contradict the behavioral game-theory models but accord well with the unique equilibrium.

Follow up experiments exploit the fact that there can be multiple equilibria in games with a unique pure-strategy Nash equilibrium. In particular, the experiment employs variations of games with a unique pure-strategy Nash equilibrium but several equilibria. The belief and choice data reveal the resulting coordination problems that play no role in traditional fixed-point theories such as Nash and QRE (which both yield unique predictions in these games).

1.1 Prior Approaches

A distinct feature of equilibrium is that beliefs and choices play a dual role. Choices satisfy an ordinal monotonicity condition, which determines the largest set of choices supported by a certain belief, and beliefs satisfy an ordinal consistency condition, which determines the largest set of beliefs that support a certain choice. In particular, -equilibrium beliefs may differ across players and may differ from -equilibrium choices. In other words, equilibrium allows for “surprises” unlike Nash and QRE, which assume correct beliefs.

1.1.1 Quantal Response Equilibrium

McKelvey and Palfrey’s (1995) Quantal Response Equilibrium (QRE) incorporates the possibility of errors into an equilibrium framework. In particular, the Nash best-response correspondences are replaced by smooth and increasing response functions, known as the “quantal response” or “better response” functions. This is the “QR” part. In addition, QRE retains the Nash-equilibrium assumption that beliefs are correct so that choice probabilities are derived relative to the true expected payoffs taking into account others’ error-prone behavior. This is the “E” part.

, ,
, ,
Table 2: A simple dominance-solvable game.

A prominent behavioral economist once quipped “I like the Q and the R but not the E.” To illustrate why QRE’s equilibrium assumption is problematic, consider the dominance-solvable game in Table 2. If denotes the probability with which Column (Row) chooses () then

in the unique Nash equilibrium. Logit QRE choice and belief probabilities, in contrast, follow from

with and non-negative “precision” or “rationality” parameters determining the sensitivity of choices with respect to expected payoffs. Note that the choice probabilities on the left are the same as those used to compute expected payoffs on the right. In other words, QRE beliefs (and choices) are fixed-points of a set of non-transcendental equations.

If QRE is seen as a model of boundedly-rational players who make mistakes it seems inconsistent to assume they are able to solve the above fixed-point equations. Indeed, not even fully rational players can solve these equations, which admit approximate numerical solutions only (and only after picking specific values for and ). Second, while players’ choices may differ in equilibrium, players must have identical beliefs about the likelihood of each choice, i.e. all players’ beliefs must coincide with a single (fixed) point in the probability simplex. This level of homogeneity in beliefs is unrealistic and will be falsified by any experiment that elicits beliefs as elements of the simplex, see Section 5.

Importantly, the “E” in QRE is far more demanding than the equilibrium assumption underlying Nash equilibrium. Not only does it involve solving transcendental equations rather than simply checking for consistency (of the type “if Row chooses then Column chooses , and if Column chooses then Row chooses ”), beliefs also have to be extremely precise. Any small change in beliefs results in a different choice profile, causing the QRE fixed-point conditions to be violated. In contrast, the Nash equilibrium is often robust to small, and possibly heterogenous, variations in beliefs. For the game in Table 2, for instance, Row’s and Column’s best responses coincide with Nash-equilibrium choices for any beliefs they hold. In other words, even when Row and Column have different and incorrect beliefs about what choices may transpire, their best responses to their beliefs support the Nash equilibrium in this game.

We are not suggesting that the Nash equilibrium is a more robust or a more reliable predictor of behavior than QRE. For that, its degenerate predictions (in case of a pure-strategy Nash equilibrium) are too easily falsified by a single deviant choice. QRE avoids such zero-likelihood problems by offering a statistical theory of behavior in games. But QRE is too demanding about the “E” part by insisting that beliefs are homogeneous – even when players have different rationality parameters – and correct – even when that requires solving transcendental equations. While the “QR” part avoids zero-likelihood problems when QRE is applied to choice data, its “E” part will surely be rejected by any data on beliefs (see Section 5).

Approaches that do allow for surprises can generally be divided into “equilibrium” or “non-equilibrium” models.

1.1.2 Equilibrium Belief-Based Models

An early example of an equilibrium model of surprises is Random Belief Equilibrium (Friedman and Mezzetti, 2005). In this model, players’ beliefs are draws from a distribution around a central strategy profile, called the “focus.” Players best respond to their beliefs and the equilibrium condition is that expected choices coincide with the focus of the belief distributions.

Rogers, Palfrey, and Camerer (2008) introduce a family of equilibrium models where player ’s choice probabilities follow from a logit quantal response function with rationality parameter . In the most general formulation, called Subjective Quantal Response Equilibrium (SQRE), players have subjective beliefs about the distributions from which others’ rationality parameters are drawn. The (Bayes-Nash) equilibrium condition is on players’ strategies, i.e. how rationality parameters map into choices, rather than on the choices themselves.

This general model can be restricted in two possible ways to establish a connection with related models. First, in Truncated Quantal Response Equilibrium (TQRE), players have downward-looking beliefs about others’ rationality parameters. Rogers, Palfrey, and Camerer (2008) show that the (non-equilibrium) Cognitive Hierarchy model can be seen as a limit case of discretized TQRE. Second, in Heterogeneous Quantal Response Equilibrium (HQRE), players have common and correct beliefs about the distributions of the rationality parameters. In this case, the equilibrium condition is on choices as in standard QRE (which arises as a limit case when the rationality-parameter distributions are degenerate).

1.1.3 Non-Equilibrium Belief-Based Models

Prominent examples include the level- model (Stahl and Wilson, 1994, 1995; Nagel, 1995) and the related Cognitive Hierarchy model (Camerer, Ho and Chong, 2004). In these models, players differ in skill and their beliefs about others’ skill levels are “downward looking.” In the level- model, level-0 randomizes or makes a “non-strategic” choice (if such a choice is easily identified). Level-1 players best respond to level-0 choices, level-2 players best respond to level-1 choices, etc. In the Cognitive Hierarchy model, level-0 randomizes while level- players, with

, assume others’ skill levels follow a truncated Poisson distribution over

and best respond to the implied distribution of choices.

Noisy Introspection (Goeree and Holt, 2004) is also a non-equilibrium model, but it does not assume downward-looking beliefs. Instead, it is based on the idea that forming higher-order beliefs is increasingly difficult.444Goeree and Holt (2004) show that Noisy Introspection can be interpreted as a stochastic version of rationalizability (Bernheim, 1984; Pearce, 1984).

1.1.4 Differences with Equilibrium

equilibrium differs from these prior approaches in several ways. First, equilibrium is a set-valued, rather than a fixed-point, solution concept. As a result, it is typically simpler to compute. Second, equilibrium is a parameter-free

theory, unlike the aforementioned models that require specific parametric assumptions for the quantal responses (e.g. logit-based SQRE), the belief distributions (e.g. Dirichlet-based RBE), the distribution of levels (e.g. Poisson-based CH), etc. As a result, its predictions can be confronted with (experimental) data without the need to estimate parameters. Third,

equilibrium treats beliefs in a way that does not neatly fit the “equilibrium” versus “non-equilibrium” classification. Unlike level- type models, equilibrium does not make ad hoc assumptions about the belief-formation process to arrive at a specific model for disequilibrium beliefs (whether it be in terms of others’ “levels of strategic thinking,” as in level- and Cognitive Hierarchy, or in terms of others’ “rationality parameters,” as in SQRE and its descendants). And unlike RBE or QRE type models, equilibrium does not assume correct beliefs. Rather equilibrium includes all beliefs that yield the same ranking of expected payoffs. The rationale is that all those beliefs support the same set of choices, and, in this sense, they sustain an equilibrium situation in which there is “no need for change.”

1.2 Organization

The next section introduces the rank correspondence, which replaces the best-response correspondence and is used to implement the ordinal monotonicity condition that options with higher expected payoffs are chosen more likely without specifying by how much. Stated differently, choice probabilities are ranked the same as expected payoffs, a requirement sometimes referred to as “stochastic rationality.” Section 3 pairs the ordinal monotonicity condition for choices with the ordinal consistency condition for beliefs to define an equilibrium. Existence for general normal-form games is proven and several properties and examples are discussed. Section 4 introduces a parametric class of -Equilibrium models where choices follow from fixed-point conditions. We show that equilibrium “minimally envelopes” these parametric models. Section 5 reports results from various experiments designed to contrast -equilibrium predictions with those of the existing behavioral game-theory models. Section 6 offers a summary of our results, and the concluding Section 7 discusses several extensions of the basic -equilibrium approach and discusses the value of equilibrium when its predictions accord well with the data and when they do not. The Appendices contain the proofs not shown in the main text, results from statistical tests based on the experimental data, as well as additional details about the methods used in the data analysis.

2 Preliminaries

Consider a finite normal-form game , where is the set of players, and for , is player ’s choice set and , with , is player ’s payoff function. Let

denote the set of probability distributions over

. An element is player ’s choice profile, which is a mapping from to , where is the probability that player chooses . Player ’s beliefs about player are represented by , which is a mapping from to , where is the probability that assigns to player choosing .555Including player ’s beliefs about her own choices in is done for notational convenience only. Throughout we assume player ’s belief about her own choices are correct, i.e. for . The concatenation of the is player ’s belief profile, , which is a mapping from to . Player ’s expected payoff given belief is where . Choosing for sure is represented by the choice profile and the associated expected payoff, given belief (with ), is denoted . Finally, represents all of player ’s “pure choices” and

denotes the vector of associated expected payoffs.

Figure 2: The image of the correspondence (top panel) and the correspondence (bottom panel) for different . If there are no ties between elements in then both and return a single point in the simplex. The three (six) possible such cases are depicted in the left panels. In case of exactly one tie, yields one of the three edges of the simplex while yields one of six line segments in the simplex, as show by the middle panels. Finally, if all elements in are equal, then equals the entire simplex while is equal to a hexagon in the simplex’ interior returns, see the right panels.

A player’s best-response correspondence maps the vector of expected payoffs to a player’s choice profile. This mapping may be point or set valued, e.g. the best-response correspondence assigns probability 1 to the choice with the strictly highest payoff but it allows for mixtures when there is a tie for the highest payoff. Player ’s best-response correspondence can generally be defined as the convex hull of the elements in that yield the highest expected payoff.

(1)

For the case of possible actions, the best-response correspondence is illustrated in the top panel of Figure 2. The left-most simplex pertains to the case when there is a strict highest payoff in which case the best-response is one of the vertices of the simplex. In the middle panel, two options tie for the highest payoff and now the best-response correspondence produces an edge of the simplex. Finally, in the right panel all three options have the same payoff and the best-response correspondence is the entire simplex.

To allow for the possibility of suboptimal choices, we soften the payoff maximization rule that underlies the best-response correspondence (as in QRE) but in a manner that retains its ordinal nature (unlike QRE). Let and let denote the set of vectors that result by permuting all elements of . Then player ’s rank correspondence is defined as:

(2)

The bottom panel of Figure 2 illustrates the rank correspondence for the case . If there is no tie the rank correspondence yields one of the six points in the left panel, when two payoffs tie the rank correspondence produces one of the six line segments in the middle panel, and when all three payoffs tie the rank correspondence produces the hexagon in the right panel.

The best-response and rank correspondences share some important features. First, their images are closed and convex sets (see e.g. Figure 2), i.e. both are upper-hemicontinuous correspondences. Second, both define idempotent mappings in the sense that and . Consider, for instance, the two-dimensional case: when , when , and when . Note that , , and

where the middle term occurs when . Hence, for all . A similar argument establishes for all . This result is intuitive: ranking alternatives that were already ranked results in the same outcome.

The best-response and rank correspondences also differ in important ways. First, the image of the rank correspondence is contained in the interior of the simplex, i.e. all options are assigned strictly positive probability, while the best-response correspondence may assign zero probability to one or more options. Second, non-optimal options matter for the rank correspondence but not for the best-response correspondence (which is why there are only three vertices in the top-left simplex compared to six vertices in the lower-left simplex of Figure 2). Stated differently, ranking options retains all ordinal information about their expected payoffs, while some information is lost when picking only the best. As a result, , but not necessarily .

3 Equilibrium

Let and denote the concatenations of players’ choice and belief profiles respectively, and let and denote the concatenations of players’ rank correspondences and profit functions. We write for the profile of expected payoffs based on players’ beliefs and for the profile of expected payoffs when beliefs are correct, i.e. for . The set of possible choice profiles is and the set of possible belief profiles is .

Definition 1

We say form an M Equilibrium if they are the closures of the largest non-empty sets and that satisfy

(3)

for all , . The set of equilibria of is denoted .

The characterization of equilibrium in (3) provides an intuitive generalization of Nash equilibrium. The assumption of perfect maximization, , is replaced with an ordinal monotonicity condition, , and the assumption of perfect beliefs, , is replaced with an ordinal consistency condition, .

Proposition 1

is non-empty for any normal-form game.

Proof. Recall from Section 2 that the rank correspondence is upper-hemicontinuous and idempotent. Kakutani’s (1941) fixed-point theorem implies existence of a profile, , that satisfies , and since is idempotent we have . Hence, and .      

Example 1. Consider a matching-pennies game where a Row and Column player choose Heads or Tails. Row receives 1 if their choices match and loses 1 if they don’t. Column’s payoffs are the negative of Row’s payoffs. Let denote the profile where each player randomizes uniformly over Heads and Tails. Players’ expected payoffs are the same when evaluated at , which is thus a Nash-equilibrium profile. Moreover, , so the Nash-equilibrium condition, , implies the -equilibrium condition, . It is readily verified that is the only profile such that . Hence, consists of a single Nash-equilibrium profile in this (non-generic) example.

In general, , so the argument in Example 1 cannot be applied to show that any Nash equilibrium is an equilibrium. For instance, for a pure-strategy Nash-equilibrium profile in a game with three or more options, is multi-valued while is single-valued when the expected payoffs of non-optimal choices are unequal. Hence, and , so pure-strategy Nash-equilibrium profiles are generally not examples of equilibria. Instead, they arise as boundary points of the -equilibrium sets. The same is true for degenerate mixed-strategy Nash-equilibrium profiles that lie on the boundary of the simplex.666Note that for any non-degenerate mixed-strategy Nash-equilibrium profile , which are thus always part of equilibrium (see also Example 1). We next present a slightly relaxed definition of equilibrium that allows for the inclusion of these boundary cases more directly.

3.1 Semi-Algebraic Geometry of Equilibrium

Here we present an alternative definition of equilibrium that highlights its connection with semi-algebraic geometry. Recall that a semi-algebraic set is defined by a finite number of polynomial equalities and inequalities. Let denote the set of all possible permutations of players’ rank vectors. For let .777Each of the corresponds to one of the equally-sized parts of and .

Definition 2

Fix . We say form an M Equilibrium if they are the closures of the largest non-empty sets and that satisfy

(4)

for all , , , and . The set of equilibria of is .

Remark 1. Definitions 1 and 2 are equivalent except for profiles that lie on the simplex’ boundary. For these boundary profiles, Definition 2 relaxes the constraint that cannot have more equal elements than as implied by in Definition 1. As a result, also degenerate Nash profiles satisfy (4) while they generally do not satisfy (3). Non-emptiness of under Definition 2 thus follows from existence of Nash equilibrium.

, , ,
, , ,
, , ,
, , ,
, , ,
, , ,

[3mm]

Table 3: Examples of non-generic symmetric two-player games with three possible choices. For the game in the left panel, a payoff-indifference line lies on the simplex’ boundary. For the game in the right panel, a payoff-indifference line lies on one of the simplex’ diagonals.

1

[width=7mm,height=7mm,innerleftsep=2pt,innerrightsep=2pt]23

[width=7mm,height=7mm,innerleftsep=2pt,innerrightsep=2pt]23

[width=7mm,height=7mm,innerleftsep=2pt,innerrightsep=2pt]23

[3mm]

Table 4: A generic symmetric three-player game with three possible choices.
Figure 3: The left panel shows the symmetric -equilibrium choices and the right panel shows -equilibrium beliefs. The top and middle panels correspond to the non-generic games in Table 3 and the bottom panel to the generic game in Table 4. The green curves indicate -equilibrium profiles where two expected payoffs are equal, the red lines indicate -equilibrium profiles contained in the simplex’ boundary, and the grey lines indicate -equilibrium profiles contained in one of the simplex’ diagonals. The blue shaded areas correspond to interior -equilibrium profiles.

Definition 2 shows that equilibrium is generally characterized by a mixture of inequalities and equalities. If only inequalities suffice to describe the equilibrium then its choice and belief sets have full dimension. Because if inequalities hold for some choice and belief profiles then, by continuity of expected payoffs, they hold for choice and belief profiles that are sufficiently close. In general, however, semi-algebraic sets can have components of various dimensions.

Example 2. The top-left panel of Figure 3 shows the different components of the symmetric -equilibrium choice set for the game in the left panel of Table 3. The colored lines in Figure 3 correspond to cases where an equality in (4) holds, either because two expected payoffs are equal (green) or because the profile lies on the simplex’ boundary (red). The grey lines correspond to the simplex’ diagonals. For this non-generic game, there are four equilibria: two have dimension zero (the Nash-equilibrium profiles and ), one has dimension one (the closure of the set of profiles for ), and one has dimension two (the closure of the set of profiles that satisfy and ). Except for the Nash-equilibrium profile , these are also -equilibria under Definition 1. The middle-panel shows a case where the symmetric -equilibrium choice and belief sets are one dimensional and have measure zero. The bottom panel shows that the -equilibrium sets can, generically, have multiple connected components.

Proposition 2

For any normal-game , there is at least one and at most a finite number of equilibria, each of which consist of a finite number of components.

Proof. That there is at least one equilibrium was shown in Proposition 1. That there are finitely many equilibria follows from basic results in semi-algebraic geometry, see e.g. Coste (2002).      

3.2 Coloring of Equilibrium

The lower-dimensional -equilibrium components shown in the top and middle panels of Figure 3 arise when a special condition is met: a payoff-indifference line coincides with the boundary of the simplex or one of its diagonals. Games for which this occurs are non-generic in the sense that if the game is perturbed slightly then these lower-dimensional components disappear and only the sets of full dimension remain. The reason that full-dimensional sets do not disappear is that the rank correspondence is single-valued on the interior of these sets and, hence, the expected payoffs can be strictly ranked. By continuity of expected payoffs, this strict ranking remains the same if the game is slightly perturbed. Also, the expected payoffs based on the supporting beliefs must have the same unique ranks, which allows us to “color” the -equilibrium choice and belief sets by their rank vector.

Proposition 3

Generically,

  • an equilibrium is a set of positive measure in ,

  • an equilibrium is characterized or “colored” by the rank vector ,

  • Nash-equilibrium profiles and are boundary points of some -equilibrium choice set.

Example 3. To illustrate, consider the four games in Table 5. Let and denote Row’s and Column’s choice profiles, where and are the probabilities that Row and Column choose respectively. Then, for instance, when and when . Since the entries in the rank vectors add up to 1, we can characterize any equilibrium by the first entries of the players’ rank vectors. That leaves four possible equilibria that can be color coded: (red), (grey), (blue), and (yellow).

In Figure 4, the left panels show the actions sets, , and the right panels show the belief sets, , for the four games in Table 5.888To be precise, the choice sets are based on only the first entries of and , i.e. for Row and for Column (as the two entries of the ’s add up to 1). Correspondingly, the belief sets are based on Row’s belief about and Column’s belief about . This allows us to depict and in two-dimensional graphs. In the left panels, the square at indicates the uniform randomization profile, , and the disks indicate Nash equilibria. For each of the four games, the beliefs sets are shown in the same color as the actions they support. For example, in the symmetric coordination game, has a higher expected payoff only when a player believes the chance the other plays exceeds . The requirement then implies as shown by the same-sized yellow squares in the top panels of Figure 4. In contrast, has a higher expected payoff when a player believes the chance the other plays does not exceed . Now, the requirement implies and as shown by the red squares in the top-left and top-right panels of Figure 4 respectively.

The equilibria for the asymmetric game of chicken and the asymmetric matching-pennies game can be worked out similarly. In the game on the far-right in Table 5, is the weakly dominant choice for Row and is the dominant choice for Column. The Nash equilibria are and and the trembling-hand-perfect equilibrium is and . In this non-generic example, there is an -equilibrium of full dimension, i.e. and , and a non-colorable -equilibrium of lower dimension, i.e. .

, ,
, ,
, ,
, ,
, ,
, ,
, ,
, ,
Table 5: A symmetric coordination game, an asymmetric game of chicken, an asymmetric matching-pennies game, and a non-generic game with a continuum of Nash equilibria.
Figure 4: The left panels show the -equilibrium choice sets and the right panels show the -equilibrium belief sets for the four games in Table 5. The belief sets are shown in the same color as the actions they support. In the left panels, the square at the center indicates the uniform randomization profile and the disks / line indicate Nash equilibria.

Example 4. Next, consider the three symmetric games in Table 6. The left-most game has a unique mixed-strategy Nash equilibrium, the middle game is dominance solvable and has a unique pure-strategy Nash equilibrium, while the right-most game has the maximal number (7) of symmetric Nash equilibria (three pure strategy and four mixed-strategy equilibria). The corresponding -equilibrium sets are shown in Figure 5.

, , ,
, , ,
, , ,
, , ,
, , ,
, , ,
, , ,
, , ,
, , ,

[3mm]

Table 6: A symmetric game with a unique symmetric mixed-strategy Nash equilibrium (left panel), a unique symmetric pure-strategy equilibrium (middle panel), and the maximum number of symmetric equilibra (right panel).
Figure 5: The left panels show the -equilibrium choice sets and the right panels show the -equilibrium belief sets for the three games in Table 6. The belief sets are shown in the same color as the actions they support. In the left panels, the square at the center indicates the uniform randomization profile and the disks indicate Nash equilibria.

Different from the case, the symmetric -equilibrium choice sets for the case may contain neither a Nash-equilibrium profile nor the the random-behavior profile. For example, the bottom panel shows a case of a completely disconnected set for which this is the case.999We will discuss this possibility in more detail in the next section where we compare equilibrium with parametric theories such as QRE. The middle panels show a case where both players choosing “yellow” is the unique pure-strategy Nash equilibrium. Most of the belief set is “blue,” however, and these beliefs support choice profiles where “blue”’ is the most likely choice.

Proposition 4

Generically,

  • may be even or odd, and may be less than, equal, or greater than the number of Nash equilibria.

  • An -equilibrium may contain zero, one, or multiple Nash equilibria.

  • The measure of an -equilibrium choice set is bounded by

  • In contrast, an -equilibrium belief set may have full measure.

Proof. Properties (i)-(ii) are demonstrated by Figures 4 and 5. Property (iv) holds, for instance, for any dominance-solvable game. Property (iii) follows since each can be partitioned into equally-sized subsets indexed by the ranks of the entries of the choice profiles it contains. Since must be constant on an -equilibrium choice set for all , the -equilibrium choice set must be contained in the Cartesian product of a single such subset for each player. Hence, its size cannot be larger than      

A priori, the set-valued nature of equilibrium might have been considered a drawback as it might render its predictions non-falsifiable. Proposition 4 shows this is not the case. The size of an -equilibrium choice set falls quickly (in fact, factorially or exponentially fast) with the number of players and the number of possible choices.

Our interest in colorable -equilibria is based on the intuition that they are the empirically relevant ones. To make this precise, we define a new stability notion called behavioral stability. For , let denote the set of normal-form games that result from by perturbing any of its payoff numbers by at most .

Definition 3

We say that is a behaviorally stable profile of the normal-form game if there exists an such that for all . Let denote the closure of the set of behaviorally stable profiles of .

In words, a choice-belief profile is behaviorally stable for the game if it is -equilibrium profile for as well as for all nearby games. Note that behavioral stability is a sharpening of the “strategic stability” criterion introduced in Kohlberg and Mertens (1986). The latter requires a strategically-stable Nash profile to be “close to” the Nash equilibrium of the perturbed game. Behavioral stability, in contrast, requires the -equilibrium profile to be an -equilibrium of the perturbed game.

Proposition 5

If is colorable then .

Proof. Profiles in the interior of a colorable -equilibrium are those for which the rank correspondence is single valued, i.e. that can be characterized by a single vector . This means that at those profiles, expected payoffs can be strictly ranked. Since expected payoffs are continuous in the payoff numbers, they will be ranked the same for games that are sufficiently close. Hence, the interior of is contained in , and since is closed, .      

Remark 2. Generically, only colorable -equilibrium profiles are behaviorally stable. For instance, for the game in the left panel of Table 3 only the profiles in the full-dimensional set in the top panel of Figure 3 are behaviorally stable. However, it is readily verified that for the matching-pennies game of Example 1, the behaviorally stable set consists of a single non-colorable -equilibrium profile.

4 Parametric Models of Stochastic Choice

Like equilibrium, the parametric models introduced in this section obey the ordinal monotonicity condition that choice probabilities are ordered the same as their associated expected payoffs. Unlike set-valued equilibrium, however, their predictions are based on fixed-points. For , let and let denote the set of vectors that result by permuting the elements of . For , we define player ’s correspondence as follows

(5)

The correspondence includes random behavior and best response behavior as special cases and satisfies a generalized idempotence condition.

Proposition 6

For all ,

  • for all when .

  • for any .

  • for any .

  • for any such that is single valued.

  • for any .

For , let denote the concatenation of the for .

Definition 4

We say is a -Equilibrium of the normal-form game if, for all ,

(6)

and is the closure of the largest set, , such that for all . The set of all -equilibria of is denoted .

The image of the correspondence is a closed and convex set (see e.g. Figure 2 for the case of and ). Existence of a -equilibrium thus follows from Kakutani’s (1941) fixed-point theorem. This also implies that is non-empty as if .

Proposition 7

is non-empty for any and any normal-form game .

While -equilibrium choice profiles are defined as fixed-points, they are easy to compute. The reason is that the right side of (6) does not vary continuously with players’ beliefs but, instead, is a piecewise-constant function over the simplex that takes on only finitely many values.

Example 5. To illustrate the simplicity of -equilibrium computations, consider the asymmetric game of chicken in the second panel of Table 5. Let and where () denotes the probability with which Column (Row) chooses . To obtain a parsimonious model, let where is a “rationality parameter.” Then the responses are

Figure 6: Row’s (solid) and Column’s (dashed) ordinal responses for the asymmetric game of chicken in the second panel of Table 5 when as indicated in the panels.
Figure 7: In the left panel, the colored (dashed) curves show the -equilibrium (logit-QRE) choice correspondence for the asymmetric game of chicken in the second panel of Table 5 for . The square at the center corresponds to random behavior () and the disks correspond to Nash equilibria (). In the right panel, the colored areas correspond to the supporting beliefs, , for the different ranges of .

The responses are “flat,” i.e. , when , and they limit to standard best responses when . Figure 6 shows Row’s and Column’s responses when increases from 0 in the top-left panel to 5 in the bottom-right panel. The intersection of the correspondences typically consists of an odd number of points (1 or 3) except at , in which case it contains and any , and at , when a bifurcation occurs and the intersection contains and any .

More generally, let denote the -equilibrium correspondence, which consists of a choice part and an associated supporting belief part: . The colored curves in the left panel of Figure 7 show the -equilibrium correspondence while the dashed curves show the logit-QRE correspondence. There are some similarities. Each have a “principal branch” that starts at the center when and ends at the pure-strategy Nash equilibria when . And each have an additional branch that connects the other pure-strategy Nash equilibrium, , with the mixed equilibrium, .

There are also some differences. First, the -equilibrium correspondence can be computed easily and characterized analytically.101010The full description of the -equilibrium correspondence for the asymmetric game of chicken in the second panel of Table 5 is:

where the first part corresponds to actions and the second part to beliefs about the other player. In the above expressions, denotes a point and an interval. More importantly, -equilibrium choices are generally supported by a range of beliefs. In the right panel of Figure 7, the three differently colored rectangles show the supporting beliefs that correspond to the part of the -equilibrium correspondence with the same color in the left panel. In contrast, the beliefs that support logit-equilibrium choices are forced to be correct and, hence, have zero measure, see the dashed curves in the right panel of Figure 7.

We next generalize the findings in Example 5. For this we need to define the two possible limit cases, i.e. and , more generally. The former case corresponds to random behavior, which leads us to define: . The latter case corresponds to best-response behavior, which leads to the following definition.

Definition 5

We say form a belief-augmented Nash equilibrium (BEAUNE) if is a best response to any belief in , i.e. for all ,

(7)

and is the closure of the largest set, , such that for all .

For instance, for the game in Table 2 of the Introduction, contains all possible beliefs, reflecting the dominant-strategy nature of the game. In contrast, the non-degenerate mixed-strategy Nash-equilibrium profile for the asymmetric matching-pennies game in Table 5 is supported only by the profile itself.

Proposition 8

For , define where , and let denote their concatenation. The -equilibrium correspondence

has the following properties for generic games:

  • is upper hemicontinuous.

  • is odd and has strictly positive measure for almost all .

  • limits to a BEAUNE when and to RAND when .

  • has a principal branch that connects RAND to exactly one BEAUNE. The other BEAUNE are connected as pairs.

Remark 3. These properties are standard and mimic those of the logit-QRE correspondence, see McKelvey and Palfrey (1995, 1996). They point out that not all Nash equilibria arise as limit points of QRE, and they call those that do “approachable.” Likewise, not all BEAUNE arise as limit points of when . We will see, however, that colorable BEAUNE, i.e. those that are boundary points of a colorable -equilibrium, are approachable.

The main interest in Proposition 8 is that it offers a single-parameter alternative to logit-QRE, which does not restrict beliefs to be correct. It would be more realistic, however, to assume that players’ rationality parameters differ, which begs the question what choices and beliefs occur under a heterogeneous -equilibrium. We next show that by varying , the -equilibrium models “fill out” the set of equilibria.

4.1 Equilibrium as a Meta Theory

We first compare equilibrium with the Luce-QRE model for which analytical solutions exist for simple games.

Example 6. To glean some intuition, consider the asymmetric matching-pennies game in the third panel of Table 5. Suppose players’ choice probabilities follow from expected payoffs using a Luce choice rule, e.g. the chance that Column and Row choose is