Evolution of preferences under perfect observability: almost anything is stable Florian Herold∗and Christoph Kuzmics† This Version: February 27, 2008
Abstract We study the evolution of preference under perfect and almost perfect observability in symmetric 2-player games as in Dekel, Ely, and Yilankaya (Review of Economic Studies, forthcoming). Dekel, Ely, and Yilankaya show that under perfect or almost perfect observability only efficient outcomes can be stable. They allow for all von NeumannMorgenstern preferences over outcomes. We demonstrate that if we allow for even more general preferences, i.e. preferences that may depend also on the opponent’s type, then any symmetric outcome with associated payoffs above the minmax value can be sustained by evolutionary stable preferences under perfect and almost perfect observability.
Keywords: Evolution of preferences, observability, discrimination, stability JEL codes: C72, C73 ∗ Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University,
[email protected] † Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University,
[email protected].
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1
Introduction
The literature on the evolution of preferences following the ”indirect evolutionary approach” by G¨ uth and Yaari (1992) and G¨ uth (1995) provides two sets of results. On the one hand evolution can favor certain non-materialistic preferences such as altruistic, spiteful, or reciprocal preferences if players observe their opponents preferences at least to some degree. For results of this nature see e.g. Bester and G¨ uth (1998), Ko¸ckesen, Ok, and Sethi (2000b), Ko¸ckesen, Ok, and Sethi (2000a), and Sethi and Somanathan (2001). On the other hand if opponents’ preferences are not observable (and players are selected randomly from a large population) evolutionary forces favor preferences, which coincide with the material payoff (evolutionary fitness) (see e.g. Ok and Vega-Redondo (2001)) or at least evolution leads to equilibrium play ’as if’ players where purely motivated by their fitness (see e.g. Ely and Yilankaya (2001)).1 Dekel, Ely, and Yilankaya (2007) make several important contributions to the literature on the evolution of preferences. First, they point out that for most of the results of the first kind another important aspect is the modeler’s restriction to certain subclasses of preferences nature can choose from. Dekel, Ely, and Yilankaya then proceed to study all symmetric 2player games and allow nature to choose from all von-Neumann Morgenstern preferences over outcomes. They also highlight the importance of the assumptions on observability: if preferences are not observable (or almost unobservable) stable outcomes need to be a Nash equilibrium, whereas if preferences are perfectly (or almost perfectly) observable a stable outcome must be efficient in their setting. So they identify efficiency as the driving force for the evolution of preferences under perfect and almost perfect observability. In this note we show that this result for perfect and almost perfect observability changes drastically if we extend the set of possible preferences even further: almost any outcome becomes stable. For the case of perfect or almost perfect observability, i.e. the case where a player with probability almost one will observe the other player’s preference type, it seems natural that preferences may depend not only on the outcome of the game but also directly on the opponent’s type itself. This is the only departure 1 If players are selected from small populations or live in small groups certain nonmaterialistic preferences may again survive, see e.g. Huck and Oechsler (1999) or Herold (2003).
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we make from Dekel, Ely, and Yilankaya (2007) in this paper. If we allow nature to choose preferences from this larger class we find that under perfect and almost perfect observability any symmetric outcome above the minmax material payoff can be sustained by evolutionary stable preferences. The key force leading to our result is discrimination, i.e. preferences that discriminate against other preference types. The intuition for our result is the following. Consider a monomorphic population of individuals who all have the same preference type and play the same strategy such that this behavior is a subjective equilibrium given individuals’ preference type and such that their material payoff is above the minmax value. Suppose furthermore that this preference type has ”spiteful” preferences over outcomes if matched not with its own type but some other type. Spiteful in the sense that the individual’s aim is to achieve the lowest possible material payoff for any type other than its own. Now suppose a small fraction of mutant preference types enter the population. Given the high degree of observability this mutant will typically be recognized as being a different type and will, hence, be minmaxed by the incumbent type. The incumbent will suffer only a little in terms of material payoff. When meeting and recognizing its own type the incumbent will continue to play the original strategy and hence do strictly better than any mutant can do. To make our argument precise we need the following ingredients. First, we need to define preferences over preference types in a consistent way. Our definitions are inspired by a paper on voluntary commitment devices by Kalai, Kalai, Lehrer, and Samet (2007). Second, for every (symmetric) outcome we find preferences (over outcomes) for which this outcome is the unique symmetric Nash equilibrium when playing against their own type. We shall call these preferences generalized Hawk-Dove preferences. Third, we find (spiteful) preferences (over outcomes) that make sure that no other preference type can obtain significantly more than his minmax material payoff (in expectation). Finally, we define a fully discriminating preference type as one who has generalized Hawk-Dove preferences when facing her own type but has spiteful preference when facing any other type. We show that this preference type is stable if preferences are observed with a sufficiently high probability.
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2 2.1
The model The environment
We will use notation as closely as possible to that in Dekel, Ely, and Yilankaya (2007), hereafter DEY, to facilitate a comparison. Let G be a symmetric 2-player game with finite action set A = {a1 , ..., an } and (material) payoff function π : A × A → IR, which can be extended (by taking expectations) to the set of all mixed strategies ∆. Without loss of generality we will assume that payoffs π are between 0 and 1. Let M denote the matrix of material payoffs, with entries all in [0, 1]. I.e. for all σ, τ ∈ ∆ we have that π(σ, τ ) = σM τ . We introduce this matrix of material payoffs here as it will be useful when we define certain preference types later. These material payoffs π represent fitness or evolutionary success and regulate the future occurrence of each preference type. Players can differ in their (subjective) preferences over outcomes. In particular, subjective preferences may differ from the material payoffs. Preferences determine players’ strategies, strategies in turn determine outcomes, the material payoffs of each type, and thereby the evolutionary success. A preference type in DEY is a function over outcomes in A × A into the real line. The set of all such preference types can be represented by the set 2 [0, 1]n (modulus affine transformations). Here we make our key departure from DEY. We extend their set of preference types and allow preferences to depend additionally on the opponent’s preference type. The following approach of modelling preferences that condition on the opponent’s preferences avoids any potential inconsistencies. It was inspired by a model of conditional commitment devices by Kalai, Kalai, Lehrer, and Samet (2007). Let Θ be a set of types. At this point this can be anything. Now consider a function u : Θ × Θ × A × A → [0, 1]. Again, at this point this can be any function. This function u induces a function uθ : Θ × A × A → [0, 1] for every θ ∈ Θ. We interpret uθ as the preference-function of a type θ. Note that by assuming uθ is constant in its first argument we could replicate all preference types of DEY. By allowing uθ to vary also in its first argument, we are basically assuming that types have preferences over other types, or, to be more precise, an individual’s preferences over outcomes can depend on the type of the opponent. Let U = {uθ : Θ × A × A → [0, 1]} denote the set of all preferences. If we assume, without loss of generality, that uθ 6= uθ0 for any θ, θ0 ∈ Θ, we 4
then have a bijection between Θ and U. Hence, Θ can again be thought of as the set of all preference types. Note that we could also assume, as in von Widekind (2004), that preferences are not necessarily of expected utility form, i.e. we could have u : Θ × Θ × ∆ × ∆ → [0, 1]. This would generate even more preferences, but would not change our main results (see section 4). Definition 1 ((Preference Space)) A space of preferences of G is a pair (Θ, u : Θ × Θ × A × A → [0, 1]). Θ is a nonempty set of possible preference types. We interpret uθ1 (θ2 , a1 , a2 ) ≡ u(θ1 , θ2 , a1 , a2 ) ∈ [0, 1] as the subjective utility of a player of type θ1 playing a1 if he plays against an opponent of type θ2 who plays a2 . Finally, as in DEY, individuals observe the opponent’s type (perfectly) with probability p ∈ [0, 1], while with remaining probability 1 − p an individual observes the completely uninformative signal φ.
2.2
The solution concept
The point of this paper is to show that with a rich enough set of preference types any symmetric outcome above the minmax material payoff is stable. Hence, using a more demanding stability concept strengthens our results. We use an extremely demanding stability concept, that we call strong stability. In particular, strong stability of an outcome (as defined below) implies stability of that outcome according to the definition of DEY. Conveniently, many things will simplify. Let P(Θ) denote the set of all finite support probability distributions on Θ.2 Let µ ∈ P(Θ). Let, as in DEY, Γp (µ) denote the Bayesian game in which nature first draws two types independently according to µ and then each individual independently observes the other’s type with probability p ∈ [0, 1], while with probability 1 − p a player observes the uninformative signal φ. Let Γ(µ) denote the complete information game corresponding to p = 1. A strategy for preference type θ is a function bθ : C(µ) ∪ {φ} → ∆, where C(µ) denotes the support of µ. Let uθ (θ0 , σ, bθ0 ) denote the expected 2
We conjecture that we do not necessarily have to restrict attention to distributions over types with finite support.
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subjective utility a player with preference type θ receives when playing mixed strategy σ ∈ ∆ against the observed type θ0 who plays bθ0 . Let b denote the profile of all bθ -functions. The profile b is an equilibrium profile if, for every θ, θ0 ∈ C(µ): ¡
¡
¢
¡
¢¢
bθ (θ0 ) ∈ argmaxσ∈∆ puθ θ0 , σ, bθ0 (θ) + (1 − p)uθ θ0 , σ, bθ0 (φ)
,
and £
¤
bθ (φ) ∈ argmaxσ∈∆ IEθ0 ∼µ puθ (θ0 , σ, bθ0 (θ)) + (1 − p)uθ (θ0 , σ, bθ0 (φ)) . Let Bp (µ) denote the set of all such equilibrium profiles in Γp (µ). Let Πθ (µ|b) denote the expected material fitness of preference type θ ∈ C(µ) given the distribution of types µ and the equilibrium profile b ∈ Bp (µ), i.e., as in DEY, X
Πθ (µ|b) =
h
¡
¢
¡
¢
p2 π bθ (θ0 ), bθ0 (θ) + p(1 − p)π bθ (θ0 ), bθ0 (φ)
θ 0 ∈C(µ)
(1)
i
p(1 − p)π (bθ (φ), bθ0 (θ)) + (1 − p)2 π (bθ (φ), bθ0 (φ)) µ(θ0 ),
+
or in matrix notation Πθ (µ|b) =
h
X
p2 bθ (θ0 )M bθ0 (θ) + p(1 − p)bθ (θ0 )M bθ0 (φ)
θ0 ∈C(µ)
+
i
p(1 − p)bθ (φ)M bθ0 (θ) + (1 − p)2 bθ (φ)M bθ0 (φ) µ(θ0 ).
Let µ ∈ P(Θ) be the incumbent preference distribution. Let µ0 ∈ P(Θ) be a distribution over entering mutant preferences. Suppose that altogether this µ0 distribution invades with a small fraction ² > 0. The post-entry mix of preference distribution is then given by µ ˜² = (1 − ²)µ + ²µ0 . For a given configuration (µ, b), a parameter δ, and a post-entry population µ ˜ the set of nearby equilibria is given by n
¯ ³
´
¯
o
¯ ¯ Bpδ (˜ µ|b) = ˜b ∈ Bp (˜ µ) : ¯x ˜b, µ ˜ − x (b, µ)¯ < δ .
For a configuration (µ, b) let, as in DEY, x(µ, b) be the induced probability distribution over actions A × A. 6
Definition 2 (Strong stability) A configuration (µ, b) is strongly stable if there exists a preference type θ ∈ Θ such that 1. µ = µθ , where µθ is the Dirac distribution on θ, and b is the unique equilibrium of Γp (µθ ), i.e. Bp (µθ ) = {b}. 2. for every δ > 0 there is an ²¯ > 0 such that for every ² ∈ (0, ²¯) and for every µ0 ∈ P(Θ) we have Bp (˜ µ² ) = Bpδ (˜ µ² |b) 6= ∅ and Πθ (˜ µ² |b0 ) > Πθ0 (˜ µ² |b0 ) for every θ0 6= θ with θ0 ∈ C(µ0 ) and for every b0 ∈ Bp (˜ µ² ), where µ ˜² = (1 − ²)µθ + ²µ0 . An outcome x is strongly stable if there exists strongly stable configuration with that outcome, i.e., there exists a strongly stable (µ, b) with x = x(µ, b). In words, we call an outcome and its supporting configuration strongly stable if and only if 1. it is induced by a configuration which consists of a single incumbent preference-type 2. with a strategy which is the unique equilibrium given the game induced by this single type; 3. for any small amount of entering mutant preference types there always exists an equilibrium; 4. all resulting equilibria remain nearby; 5. and in all these equilibria the incumbent preference-type receives a strictly higher material payoff than any other type in the post-entry configuration. This definition may seem too demanding. Yet we can prove our main result for strong stability. As strong stability of an outcome implies stability of an outcome in the sense of DEY this only strengthens our results. We summarize briefly the stability concept used in DEY. A detailed discussion is in their paper pp. 7-9. DEY call a configuration (µ, b) balanced if Πθ (µ|b) = Πθ0 (µ|b) for all θ, θ0 ∈ C(µ). Furthermore, ˜ = {µ0 : µ0 = (1 − ²0 )µ + ²0 θ, ˜ ²0 < ²} N² (µ, θ) 7
denotes the set of all preference distributions resulting from entry of no ˜ Then DEY define focal equilibria. more than ² mutants, all of type θ. Beginning with a configuration (µ, b), and following an entry by θ˜ leading to ˜ an equilibrium is focal if incumbents’ behavior is unchanged, µ ˜ ∈ N² (µ, θ), i.e., ˜bθ (θ0 ) = bθ (θ0 ) (whenever p > 0) and ˜bθ (∅) = bθ (∅) (whenever p < 1) for all θ, θ0 ∈ C(µ). Bp (˜ µ|b) denotes the set of all focal equilibria to b if the distribution is µ ˜. Then, DEY define stability as follows.
Definition 3 (DEY-Stability) A configuration (µ, b) is DEY-stable if it is balanced and if for every δ > 0 there exists ² > 0 such that for every θ˜ ∈ Θ ˜ and µ ˜ ∈ N² (µ, θ), 1. Πθ (˜ µ|˜b) ≥ Πθ˜(˜ µ|˜b) for all ˜b ∈ Bp (˜ µ|b) and θ ∈ C(µ). 2. If Bp (˜ µ|b) = ∅, then Bpδ (˜ µ|b) 6= ∅ and Πθ (˜ µ|˜b) ≥ Πθ˜(˜ µ|˜b) for all ˜b ∈ δ Bp (˜ µ|b) and θ ∈ C(µ). An outcome x is DEY-stable if there exists a DEY-stable configuration with that outcome, i.e., there exists a DEY-stable (µ, b) with x = x(µ, b). A preference distribution µ is DEY-stable if (µ, b) is DEY-stable for some b.
In words, an outcome and its supporting configuration is DEY-stable if and only if 1. all preference types in the configuration receive the same material payoff; 2. the strategy profile used by the various types constitutes an equilibrium; 3. for any small amount of entering mutant preference types in all focal equilibria of the post-entry population the incumbent preference-types receive a material payoff which is at least as high as that of any other type in the post-entry configuration. 4. if there exists no focal equilibrium then there exists a nearby equilibrium, and in all these nearby equilibria the incumbent preference-types receive a material payoff which is at least as high as that of any other type in the post-entry configuration. 8
Lemma 1 below shows that every strongly stable outcome is also DEYstable. The sketch of the proof is as follows. If an outcome is strongly stable it is supported by a single preference type, hence, is trivially balanced. While DEY-stability requires the existence of at least one nearby equilibrium, while strong stability requires existence as well as that all equilibria are nearby. Finally strong stability requires that all mutants receive strictly lower material payoff in all equilibria, while DEY-stability only requires that mutants receive material payoffs which are less than or equal to the incumbent’s material payoff.
Lemma 1 If an outcome is strongly stable it is also DEY-stable.
Proof: Suppose x ∈ P(A × A) is strongly stable. Then there is a θ such that Γp (µθ ) has a unique equilibrium b with x = x(b, µθ ). We show that the configuration (µθ , b) is stable. The configuration (µθ , b) is trivially balanced (since C(µθ ) = {θ}). Let δ > 0. Choose ² = ²¯, where ²¯ > 0 is given by condition 2 of the strong stability definition. For an arbitrary θ˜ ∈ Θ ˜ we have µ and µ ˜ ∈ N² (µθ , θ) ˜ = (1 − ²0 )µθ + ²0 µθ˜ ≡ µ ˜²0 with ²0 < ²¯. From 0 condition 2 for strong stability we have Πθ (˜ µ²0 |b ) > Πθ0 (˜ µ²0 |b0 ) for every b0 ∈ Bp (˜ µ²0 ). This implies condition 1 of DEY-stability since C(µθ ) = {θ} and Bp (˜ µ²0 |b) ⊂ Bp (˜ µ²0 ). Similarly, condition 2 of strong stability implies condition 2 of DEY-stability since Bpδ (˜ µ²0 |b) = Bp (˜ µ²0 ) 6= ∅ and C(µθ ) = {θ}. QED
3
Results
Let G be a general symmetric 2-player game with finite pure strategy space A with n elements as described in section 2.1. Recall that M denotes the matrix of material payoffs. Let π ¯ denote the expected material payoff each of the two players could guarantee for him- or herself, i.e. π ¯ = maxσ∈∆ minτ ∈∆ σM τ . The following first result in this paper, Proposition 1, is provided for completeness and demonstrates the intuitive result that an outcome with material payoff below the material minmax value of π ¯ can not be sustained in even a DEY-stable configuration, and, hence, by Lemma 1, can also not be sustained in a strongly stable configuration.
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Proposition 1 Let Θ be an arbitrary preference space, except that it contains the preference type θm that independently of the opponent’s type coincides with the material payoffs, i.e. uθm (θ, σ, τ ) = σM τ . Let (µ, b) be a configuration such that Πθ (µ|b) < π ¯ for some θ ∈ C(µ). Then (µ, b) is not DEY-stable (and therefore not strongly stable) under any degree of observability p ∈ [0, 1].
Proof: Suppose (µ, b) be a configuration such that Πθ (µ|b) < π ¯ for some θ ∈ C(µ). To have DEY-stability we must have that (µ, b) is balanced, and, hence, Πθ (µ|b) < π ¯ for all θ ∈ C(µ). Now consider the preference-type θm such that uθm (θ, σ, τ ) = σM τ for all θ ∈ Θ. I.e. this type has preferences that do not depend on the opponent and coincide with the material payoffs. Now, for ² > 0 let µ ˜² = (1 − ²)µ + ²µ0 , where µ0 is the Dirac-measure on type 0 θ . For given δ > 0 let ² be small enough such that any focal equilibrium is also a nearby equilibrium, i.e. Bp (˜ µ² |b) ⊂ Bpδ (˜ µ² |b). Given the post-entry distribution of types, µ ˜² , either one of two statements must be true. Either there is no nearby equilibrium profile b0 , in which case (µ, b) is not DEYstable, or there is a nearby equilibrium profile b0 . If the latter is true then the behavior of the incumbent types cannot have changed much in the postentry population. I.e. if ² is sufficiently small we still have that Πθ (˜ µ² |b0 ) < π ¯ 0 for all θ ∈ C(µ). The mutant type θ , however, will achieve a payoff of at least Πθ0 (˜ µ² |b0 ) ≥ π ¯. QED The intuition behind this result is that a mutant whose preferences coincide with the material payoffs will always do at least as well as the minmax payoff and, hence, is able to invade successfully. In order to prove our main results the following definitions and lemma are useful. For any y ∈ (Θ), with all yi > 0, consider the following subjective payoff matrix
0
1 c(y)y1 0 Ay = 0 .. .
0
0 0 1 c(y)y2
0 .. . 0
0 ··· 0 ··· 0 ··· .. .. . . .. .. . . ··· 0
P
0 0 0 .. .
1 c(y)yn
0 0
0
0 .. .
1 c(y)yn−1
0
,
where c(y) = ni=1 y1i . This payoff matrix is such that whatever the opponent plays the other player always strictly prefers to play one strategy 10
higher, except if the opponent plays the last strategy, then the other player strictly prefers to play the first strategy. If both players share this payoff matrix then the resulting game has a unique symmetric Nash equilibrium which is given by exactly y. If n = 2 this game is the Hawk-Dove game. For n ≥ 3 one might call this a generalized Hawk-Dove game. For y 6∈ int(∆), i.e. yi = 0 for some i ∈ A, we have to modify the definition of Ay somewhat. Intuitively we just define Ay as before but only for those rows and columns i which are such that yi > 0. The rest of the matrix is then filled with zeros and occasional 1’s. To be more precise there are two cases which need separate treatment. First, let y ∈ ∆ be such that yi = 1 for some i ∈ A. Then Ay is such that Ayij = 1 for all j for i such that yi = 1 and all other Ayij ’s are equal to 0.(Then action i is a dominant strategy. Second, and without loss of generality3 , let y ∈ ∆ be such that yi = 0 for all i ≤ l and yi > 0 for all i > l for some l ≤ n − 2. P P Let c¯(y) = i:yi >0 y1i , i.e. c¯(y) = ni=l+1 y1i . Then for rows and columns l + 1 and above define Ay just as above, but replacing c(y) with c¯(y). All other rows, the first l, shall be zeros only. To then ensure uniqueness of the symmetric equilibrium y in this case we need to have at least one positive element in every one of the first l columns. Let, in fact Ayni = 1 for all 1 ≤ i ≤ l. The matrix Ay , in this case, can then be written as follows.
0 ··· 0 . .. .. .. . . 0 · ·· 0 0 ··· 0 . .. · · · ... Ay = . .. . · · · .. . .. · · · ... 0
··· 0 1 ··· 1
0 .. .
··· .. .
0 0
··· 0
1 c¯(y)yl+1
0
0
1 c¯(y)yl+2
0 .. .
0 .. .
0
0
··· ··· .. .. . . ··· ··· 0 ···
··· .. .
0 .. .
··· 0
0 1 c¯(y)yn
···
0
0
0 ··· .. .. . . .. .. . . ··· 0
0 .. .
0
0
0
0 .. .
1 c¯(y)yn−1
0
.
This defines Ay for all y ∈ ∆. The following result is about the game 3
Alternatively, one could define Ay for such y, with yi = 0 for some i and yi < 1 for all i, somewhat more complicatedly as follows: Let i0 = min{j : yj > 0} and let 1 1 . Furthermore Ayij = c¯(y)y i00 = max{j : yj > 0}. Then Ayi0 i00 = c¯(y)y if yi > 0 and i i00 y 0 0 0 j = max {j : j < i ∧ yj 0 > 0}. Finally Aji = 1 for j = min{j > i : yj 0 > 0} when i is such that yi = 0 and all remaining Ayij ’s are equal to 0.
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induced by these generalized Hawk-Dove preferences and shows that the game has a unique symmetric Nash equilibrium. Lemma 2 The symmetric 2-player game with payoff-matrix Ay , with y ∈ ∆, has a unique symmetric Nash equilibrium, which is y.
Proof: We will do the proof for the case y ∈ int(∆) only. The proof extends to all y ∈ ∆ straightforwardly. Let y ∈ int(∆) and let Ay be defined as above. Then y is obviously the only symmetric Nash equilibrium with full support as it is the only vector which equalizes the payoff for all strategies of the opponent. Suppose there is a symmetric NE z ∈ ∆ with non-full support. I.e. let zj = 0 for some j ∈ {1, ..., n}. Suppose first that j = n. Then strategy a1 , being good only against an , is strictly dominated by any mixture with full support in A \ {an }, and, hence, we must have z1 = 0. Suppose now that j < n, the only other case. But then strategy aj+1 is strictly dominated by any mixture with full support in A \ {aj }, and, hence, we must have zj+1 = 0. Iterating this argument provides us with zi = 0 for all i ∈ {1, ..., n}, which provides a contradiction. QED Let I n denote the n × n-matrix of all 1’s.
Definition 4 A fully discriminating preference type, indexed by y ∈ Θ,¡ denoted by θy , is such that uθy (θy , σ, τ ) = σAy τ and uθy (θ, σ, τ ) = ¢ t σ I − M τ for all θ ∈ Θ, θ 6= θy .
The fully discriminating preference type with index y, therefore, has the following preferences over outcomes. When facing her own type θy her preferences are of the generalized Hawk-Dove variety with subjective payoff matrix Ay as described above, and when facing any other type her preferences are spiteful with subjective payoff matrix I − M , implying that she will seek to minimize her opponent’s material payoff in this case. There is one additional preference type we will need for some of our results, the type whose preferences coincide with the material payoffs. Definition 5 The material preference type, denoted by θM is such that uθM (θ, σ, τ ) = σM τ for all θ ∈ Θ.
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3.1
Perfect observability
Proposition 2 Let Θ be an arbitrary preference space, except that it contains the fully discriminating preference type θy , as defined in Definition 4, for some y ∈ int(∆). Let p = 1. If π(y, y) > π ¯ , then the configuration (µ, b) is strongly stable, where µ is the Dirac measure putting probability 1 on θy and b is such that bθy (θy ) = y. Hence, for any y ∈ ∆ the outcome (y, y) is strongly stable (and therefore DEY-stable). Proof: First, given the preferences of the fully discriminating type θy and by Lemma 2 we have that B1 (µθ ) = {bθy }. Second, consider, without loss of generality, any µ0 ∈ P(Θ) such that θy 6∈ C(µ0 ). Let µ ˜² = (1 − ²)µ + ²µ0 . Note that we must have B1 (˜ µ² ) 6= ∅. Now consider any b0 ∈ B1 (˜ µ² ). We 0 δ 0 need to show that b ∈ B1 (˜ µ² ). Strategy b is a description of how all types behave when they meet each opponent type. Consider the following table which specifies the behavior in b0 . An entry in this table is the strategy employed by the row-type against the column-type.
θy θ0 ∈ C(µ0 )
θy y1 y3
θ0 ∈ C(µ0 ) y2 y4 .
All y i ’s are in ∆. Now what do we know about these strategies? When type θy meets its own type with perfect observability they must of course play µ² ). Given that the, by Lemma 2, unique equilibrium y 1 = y in any b0 ∈ B1 (˜ y most players in µ ˜² are of the θ -type this alone is sufficient to imply that for any δ > 0 we can find an ²¯ > 0 small enough such that for all ² ∈ (0, ²¯) we have that b0 ∈ B1δ (˜ µ² ). Note also that, as there may be multiple types 2 in C(µ), strategies y and y 3 in the above table may well be different for different mutant types θ0 ∈ C(µ0 ). We now turn to the material payoff each type in C(˜ µ² ) will receive. y Given the spiteful preferences of type θ and the fact that p = 1 we must also have in any b0 ∈ B1 (˜ µ² ) that y 2 , which depends on θ0 , is the strategy that minimizes the material payoff of type θ0 . This and the fact that material payoffs are constrained to be between 0 and 1 we then have the following table for material payoffs in the various encounters.
θy θ0 ∈ C(µ0 )
θy = π(y, y) ≤π ¯ 13
θ0 ∈ C(µ0 ) ≥0 ≤ 1.
This implies that, for any b0 ∈ B1 (˜ µ² ), Πθy (˜ µ² |b0 ) ≥ (1 − ²) · π(y, y) + (²) · 0, 0 while, again for any b ∈ B1 (˜ µ² ),and any θ 6= θy , Πθ (˜ µ² |b0 ) ≤ (1 − ²) · π ¯ + ² · 1. 0 For small enough ² > 0 we, therefore, have that Πθ (˜ µ² |b ) > Πθ0 (˜ µ² |b0 ) for 0 0 0 every θ ∈ C(µ) and every θ ∈ C(µ ) and for every b ∈ B1 (˜ µ² ). QED Given our definition of strong stability requires that an outcome be supported by a single type we can only obtain outcomes x ∈ P(A × A) which are of the symmetric product form, i.e. x = y · y, where y ∈ ∆. This derives from the simple fact that if there is only one type all players of this type must choose the same mixed strategy when playing against each other. Hence, by Proposition 1, π(y, y) ≥ π ¯ is a necessary condition for strong stability of an outcome y and by Proposition 2 π(y, y) > π ¯ is a sufficient condition for strong stability. Any outcome induced by a single type choosing mixed strategy y ∈ ∆ is strongly stable for p = 1 as long as the resulting material payoff π(y, y) exceeds the material minmax value π ¯. Note that Proposition 2 also provides preferences for which this outcome is strongly stable. We also then call these preferences strongly stable. These preferences here are such that the incumbent type has generalized Hawk-Dove preferences when her opponent is of the same type, but when her opponent is of another type she has preferences which are diametrically opposed to the material payoffs of the opponent. These are our fully discriminating preference types. Now while Propositions 1 and 2 identify all outcomes that are strongly stable for p = 1 they do not give us all preferences that are strongly stable. There are typically other preferences, which are also strongly stable. It is for instance not necessary for the incumbent to minmax its mutant opponents. It is sufficient to play something which allows mutants to achieve a higher material payoff as long as it is still below the material payoff of the incumbent.
3.2
Almost perfect observability
Proposition 3 Let Θ be an arbitrary preference space, except that it contains the fully discriminating preference type θy , as defined in Definition 4, for some y ∈ ∆ with π(y, y) > π ¯ . Then, for p close enough to 1, the configuration (µ, b) is strongly stable, where µ is the Dirac measure putting probability 1 on θy and b is such that bθy (θy ) = bθy (φ) = y. Proof: First, as in the full observability case, given the preferences of the 14
fully discriminating preference type θy Lemma 2 implies that Bp (µ) = {b}. Second, consider, without loss of generality, any µ0 ∈ P(Θ) with θy 6∈ C(µ0 ). Let µ ˜² = (1 − ²)µ + ²µ0 . We now need to characterize any b0 ∈ Bp (˜ µ² ). We need to determine b0θ : Θ ∪ {φ} → ∆ for all θ ∈ C(µ0 ) ∪ {θy }. First of all, we know there exists such a symmetric b0 ∈ Bp (˜ µ² ). We really only care about how type θy behaves in any such b0 . There are 3 components to this. How type θy behaves when meeting and observing its own type, when meeting and observing another type, and when observing φ. The subjective payoff to type θy when recognizing its own type and using strategy z ∈ ∆, while everyone else plays according to b0 , is given by ¡
¢
¡
¢
puθy θy , z, b0θy (θy ) + (1 − p)uθy θy , z, b0θy (φ) , which, by the fact that uθ is of the expected utility form, equals ¡
¢
uθy θy , z, pb0θy (θy ) + (1 − p)b0θy (φ) . Consider first the case y ∈ int(∆). If p is sufficiently close to 1, by the same argument as in the proof of Lemma 2, we cannot have that (b0θy (θy ))i = 0 for any i ∈ {1, ..., n}. Hence, the only possibility is that (b0θy (θy ))i > 0 for all i ∈ {1, ..., n}. This implies that type θy must be indifferent between all strategies in A. This implies that p (b0θy (θy ))i + (1 − p) (b0θy (φ))i = yi , yi −(1−p)(b0θy (φ))i , which, for p close to 1, is close or equivalently (b0θy (θy ))i = p to yi . This alone is sufficient to show that for any δ > 0 there is an ²¯ > 0 such that for all ² ∈ (0, ²¯) we have that any such b0 ∈ Bp (˜ µ² ) also satisfies 0 δ b ∈ Bp (˜ µ² ) for p sufficiently close to 1. This is also sufficient to show that, for p close to 1 and for small ² > 0, the material payoff of type θy is strictly above that of any other type θ ∈ µ0 for any µ0 ∈ P(Θ). To see this let y 0 = b0θy (θy ). The material payoff to type θy is bounded from below by h
i
(1 − ²) p2 π(y 0 , y 0 ) + O(1 − p) , where O(1 − p) is a term that converges to 0 as 1 − p tends to 0. This lower material payoff bound, hence, tends to π(y, y) if ² tends to 0 and p to 1. The material payoff to any other type θ is bounded from above by h
i
(1 − ²) p2 π ¯ + O(1 − p) + ², 15
where again O(1 − p) is a term that converges to 0 as 1 − p tends to 0. This upper material payoff bound, hence, tends to π ¯ if ² tends to 0 and p to 1. Hence, for p sufficiently close to 1 and ² sufficiently close to 0 we have that Πθ (˜ µ² |b0 ) > Πθ0 (˜ µ² |b0 ) for every θ ∈ C(µ) and every θ0 ∈ C(µ0 ) and for every b0 ∈ Bp (˜ µ² ). In the general case y ∈ ∆ the same arguments apply. Notice that all actions i with yi = 0 are strictly dominated (and will therefore not be played) for the type θy if facing his own type. QED
3.3
No and almost no observability
For the case of no and almost no observability DEY show that being a Nash equilibrium of G, i.e. in material payoffs, is a necessary condition for an outcome to be DEY-stable. Their argument still goes through in our setting: Under no and almost no observability being a Nash equilibrium is a necessary condition for DEY-stability (and therefore also necessary for strong stability). Intuitively, the opponent’s type is almost never observed anyway and discrimination against other types has no bite. More precisely, if an outcome is not a Nash equilibrium of G, then at least one type must with positive probability play an action that earns a material payoff strictly below the material payoff of the best response to the current play of the population. Then if the material preference type θM , as defined in Definition 5, is among the entrants, she is more successful than at least one of the incumbent types: consider the post entry population µ ˜² = (1 − ²)µ + ²µ0 , where µ0 M is the Dirac-measure on type θ and ² > 0. Either there is no nearby equilibrium profile b0 , in which case (µ, b) is not DEY-stable, or there is a nearby equilibrium profile b0 . If the latter is true then the behavior of the incumbent types cannot have changed much in the post-entry population. This implies that for ² sufficiently small the new outcome x0 (˜ µ² , b0 ) must be 0 δ close to the old outcome x(µ, b) for all equilibria b ∈ B0 (˜ µ² ). Given x(µ, b) does not constitute a Nash equilibrium in material payoffs we must have that also x0 (˜ µ² , b0 ) is not a Nash equilibrium in material payoffs. But then the behavior of θM in b0 must be a best-reply to x0 (˜ µ² , b0 ) and, hence, provide type θM with a higher material payoff than at least one of the incumbent types. DEY show also that being a strict Nash equilibrium is a sufficient condition for (DEY)-stability under no observability. This remains valid in our 16
setting. Furthermore, they provide a counterexample (DEY, Example 1) to show that under almost no observability being a strict Nash equilibrium is not sufficient for the stability of an outcome. Their example is the following game.
A B
A 6.6 5,0
B 0,5 2,2
(B, B) is a strict Nash equilibrium, yet not stable. They show this by defining an entrant type that plays strategy A when observing his own type and strategy B otherwise. There is a focal equilibrium in which this type receives strictly more than the incumbents. This game still works as a counterexample in our setting. Consider an incumbent population that plays (B, B). An entrant that has A as a dominant strategy against its own type and B as a dominant strategy against any other type will destabilize this configuration. Notice that if the proportion of entrants is small enough this entrant will play strategy B when observing the empty signal. He will play A only if he observes the entrant type. While this gives him with high probability a payoff of only 0, the observed entrant receives a payoff of 5, so on average entrants get payoff of 2.5 in this match which is strictly better than the incumbents’ payoff of 2. Notice that in our setting the incumbents may change their play when observing an entrant. In the current game this will not hurt the entrant as strategy B guarantees him a material payoff of at least 2. The entrant does therefore strictly better than the incumbents. Yet, this argument depends heavily on the fact that the minmax material payoff (of 2) is equal to the material payoff in the strict Nash equilibrium. In fact in our setting under almost no observability it is a sufficient condition for DEY-stability that an outcome (a, a) is a strict Nash equilibrium and that the material payoff π(a, a) is strictly above the minmax payoff. To see this consider an incumbent population of players with fully discriminating preferences θa : against their own type they have action a as a dominant strategy and they minimize the material payoff of any other type. For a sufficiently small proportion of entrants the incumbent plays strategy B when observing his own type or the empty signal. The incumbent receives a material payoff arbitrarily close to π(a, a). The entrant obtains with a probability arbitrarily close to p not more than his minmax payoff (when observed by an incumbent) and with a probability arbitrarily close to (1−p) not more than π(a, a) (when unobserved by an incumbent). Hence, any 17
equilibrium remains nearby and any entrant gets a strictly lower payoff then the incumbent.
3.4
Partial observability
Suppose the degree of observability p is any number strictly between 0 and 1. One might ask whether the transition from only Nash equilibria being stable at small p to anything above the minmax-value being stable for large p is a continuous one or whether there is a jump at some level of p. It turns out that either of these can be the case depending on the game at hand. Suppose now that p ∈ (0, 1). We investigate strong stability. Consider a configuration (µ, b) with outcome x(µ, b), µ Dirac concentrated on some fully discriminating type θy and a mutant distribution µ0 . The following table describes behavior of the various types in the post-entry population µ ˜² = (1 − ²)µ + ²µ0 .
θy θ0 ∈ C(µ0 )
b0
θy ≈y z1
θ0 ∈ C(µ0 ) minmax z2
φ b00 ≈ y z3 ,
where zi ∈ ∆ for i = 1, 2, 3. Note that type θy ’s behavior in the pre-entry game must be the same when recognizing his/her opponent (also of type θy ) and when receiving the uninformative signal φ given that all players are of the θy -type. Now if there is small fraction of mutants this behavior cannot change too much given that by Lemma 2 the original equilibrium is unique (see the proof of Proposition 3). The above table indicates that while we know how the incumbent type will behave we do not know how the mutants will behave. We can, nevertheless, give bounds on all relevant payoffs. Consider first the payoff type θy obtains. With probability 1 − ² she is matched with her own type. Given the behavior of type θy as described above, i.e. given b0 and b00 , we then must have that the material payoff to her is approximately π(y, y), tending to π(y, y) as ² tends to zero. With probability ² she obtains material of at least 0, which is the lowest possible material payoff. Hence, the payoff to θy , as ² tends to zero, is given by Πθy = π(y, y). 18
Now consider a mutant type θ0 ∈ C(µ). Now suppose this type is recognized by its opponent, which happens with probability p. Given this the probability that the opponent’s type is θy is 1−². In this case type θ0 will be minmaxed and can at most obtain a material payoff of π ¯ . With remaining probability ², still conditional on being recognized, type θ0 faces any other type in C(µ) with resulting material payoff of no more than 1, the maximal achievable payoff. Suppose mutant type θ0 is not recognized by its opponent, which happens with probability 1 − p. Again with probability 1 − ² the opponent is then type θy . In this case mutant type θ0 can take material advantage of its opponent (who plays b00 close to the original y). The best material payoff type θ0 can obtain is close to π ∗ (y) = maxz∈∆ π(z, y), tending to π ∗ (y) as ² tends to zero. In the final remaining case type theta0 meets any other type with resulting material payoff of no more than 1. All in all, the payoff to any mutant type θ0 ∈ C(µ0 ), as ² tends to zero, is at best given by Πθ0 = p¯ π + (1 − p)π ∗ (y). This is the best payoff a mutant can achieve, and this payoff is also achievable by some mutants. This proves the following Proposition. Proposition 4 The configuration (µ, b), where µ is Dirac placing probability 1 on the fully discriminating type θy for some y ∈ ∆ and bθy = bφ = y, is strongly stable if π ∗ (y) − π(y, y) p> π ∗ (y) − π ¯ and not strongly stable if p
b.4 For this PD-game we have π(y, y) = (a + b − 1)y12 + (1 − 2b)y1 + b, π ¯ = b, and π ∗ (y) = b + (1 − b)y1 . This implies that (a+b−1) preference type θy is strongly stable if and only p ≥ b−y11−b . Consider 3 cases. First a+b = 1. Then θy is strongly stable if and only if b p ≥ 1−b . This implies that in this case either nothing is stable or everything is. The smaller b, the minmax value, the more readily everything is stable. As b (and hence a in this case) approaches 12 only strategy (1, 0), and its preference type, remains stable. This is due to the fact that the game has essentially no dilemma component anymore as a is just above b. Now consider the second case, where a + b < 1. I this case it is no longer true that at a certain cutoff p all types θy turn from unstable to stable. Here, types θy with lower y1 (less cooperative) are more easily stable (for lower p). The typical picture here is, as we slowly go from p = 0 to p = 1, at some positive p strategy (0, 1) becomes stable, then as p is rising, more and more (y1 , 1 − y1 ) strategies become stable until at some p strictly less than 1 all such types θy are stable. This case is illustrated in Figure 1.
Figure 1: Strongly stable strategies (left) and payoffs (right) in the prisoners’ dilemma with a = 1/2 and b = 1/4 (second case) as a function of p. The third case, where a + b > 1 is similar to the second case, but in reverse. Taking p slowly from 0 to 1 we will, for low p, have no stable preference type until we reach a strictly positive level of p at which now the preference type θy with y = (1, 0) (cooperate) becomes stable, then more 4
If a < b no strategy other than (0, 1) (defect) achieves a material payoff of at least the minmax-value, which is here π ¯ = b. Hence, in this case, by Proposition 1 there are no stable outcomes other than, perhaps, strategy (0, 1).
20
and more types become stable until we reach a p strictly less than 1 where then all such types are stable. These examples illustrate that typically both results that almost everything is stable under p = 1 and nothing is stable under p = 0 extend fully to nearby values of p, sometimes quite far.
3.4.2
The coordination game
Consider the coordination game with material payoff matrix Ã
MC = We here have π(y, y) =
3 2
³
y1 −
1 3
´2
1 0 0 21
!
.
+ 13 , π ¯ = 31 , and π ∗ (y) = y1 if y1 >
1 3
and π ∗ (y) = 12 (1 − y1 ) if y1 < 13 . This implies that preference type θy for y1 > 31 is strongly stable if and only if p > 23 (1 − y1 ), while preference type θy for y1 < 13 is strongly stable if and only if p > 3y1 . Hence, the two Nash-equilibrium types θ(0,1) and θ(1,0) are strongly stable for all levels of p, while the type y1 = 31 is unstable for all p ∈ [0, 1]. Intermediate types are stable for some levels of p. For instance preference type θy with y1 = 32 is stable as long as p > 12 , while preference type θy with y1 = 43 is stable as long as p > 83 . This is illustrated in Figure 2.
Figure 2: Strongly stable strategies (left) and payoffs (right) in the coordination game as a function of p.
21
4
Discussion
We investigated a model of preference evolution in which nature can give individuals preferences which do not only depend on outcomes (as in previous literature) but also and importantly on the opponent’s preference type directly. Our aim of this paper is to demonstrate that perfect or almost perfect observability of preferences in such models of preference evolution induces another, and all overpowering, evolutionary force in addition to the previously identified force towards efficiency as pointed out by Dekel, Ely, and Yilankaya (2007) (DEY). This force is one of discrimination. If it is present, which is the case if nature can make a type’s preferences dependent on its opponent’s type, it is stronger than the evolutionary push towards efficiency, with the consequence that sub-optimal, i.e. inefficient outcomes may well be achieved through the evolutionary process. In fact we show that any symmetric outcome above the minmax can be sustained through a strongly stable preference profile. Let us here provide our main argument using the interpretation of Robson (1990)’s secret hand-shake. First recall that in DEY this secret handshake idea is central for the argument that non-efficient outcomes cannot be stable. Suppose that in the DEY context the incumbent plays a Paretoinefficient outcome. Then a new type can come in who e.g. is indifferent over all outcomes. One equilibrium in the post-entry population is then such that the incumbent continuous to play her strategy in all meetings, while the mutant type plays this same strategy when recognizing the incumbent type but plays the Pareto-superior strategy when recognizing her own type. Given that observability is perfect or almost perfect this new type will (almost) always recognize the opponent’s type and, hence, salvage higher material payoff than the incumbent. This recognition has the flavor of the secret hand-shake. In any meeting the new type uses the secret-handshake. If she also receives the secret hand-shake back she knows the opponent is also of her own type which now leads her to play the superior strategy. If the opponent does not use the secret hand-shake she realizes that her opponent is the incumbent type and plays the incumbent’s strategy. This is an equilibrium given the mutants indifference between all strategies. In our model the mutant can also be understood to use the secret handshake and may play a superior strategy when receiving the secret hand-shake back. The key difference, though, to DEY is that in our model an incumbent 22
who observes the secret hand-shake of the opponent, does not continue to play the old strategy but, because of her discriminatory type, will react very differently by in fact spitefully minmaxing her opponent. This paper, therefore, identifies another force, that of discrimination, which in our model outweighs DEY’s force towards efficiency. Natures ability of endowing players with discriminatory powers can thus lead to substantially inferior outcomes compared to the case in which nature can only provide preferences over outcomes as in DEY. Discrimination also plays a role in Banerjee and Weibull (2000)’s analysis5 of neutrally stable strategies in symmetric 2-player games in which players before playing the game send payoff-irrelevant messages, which can be interpreted as observable traits. They specifically characterize the neutrally stable strategies for 2 × 2 coordination games. Their neutrally stable strategies somewhat resemble the preferences we use to sustain our Propositions 2 and 3. Neutrally stable strategies are such that only messages from a subset of all available messages are used, these with equal probability. Upon seeing a message within this subset an individual plays an action associated with one of the strict Nash equilibria of the stage game6 , while upon seeing any message not from this subset the individual plays the mixed strategy which minimizes the opponent’s payoff. Yet, the fact that traits in Banerjee and Weibull (2000) do not bind the player in any way leads to results very different from ours. For general games, in any Nash equilibrium of their cheap talk game players must play a Nash equilibrium of the base game conditional on any message pair. This immediately implies that all Nash equilibrium outcomes, and, hence, payoffs which are sustained by a neutrally or evolutionary stable strategy, in the cheap talk game must lie in the convex hull of the base game Nash equilibrium payoff. This is typically a much smaller set of outcomes than we obtain in our model of preference evolution. For the prisoners’ dilemma, for instance, this is the singleton outcome associated with the dominant strategy. Even in 2 × 2 coordination games results differ substantially. Consider e.g. a payoff of 2 in the “good” Nash equilibrium and 1 in the “bad” equilibrium and an infinite number of messages. Then in Banerjee and Weibull (2000) the limit set of payoffs from 5 The title of their original working paper, (Banerjee and Weibull 1993), makes this explicit: ’Evolutionary selection with discriminating players’. 6 In fact, the individual’s behavior has to be such that upon seeing the same message she sent she plays the action associated with the payoff-inferior strict Nash equilibrium, and if seeing a message different from the one she sent, but still one of the messages from the prescribed subset, she plays the action associated with payoff-superior strict Nash equilibrium.
23
neutrally stable strategies consists of 2 and all numbers 2 − n1 for positive integers n and only the efficient Nash equilibrium is evolutionary stable. In our setting, in contrast, every payoff above the minmax payoff is strongly stable. Our argument has an interesting connection to the commitment-device folk theorem by Kalai, Kalai, Lehrer, and Samet (2007). In their model players can choose a commitment device that can condition on the commitment device chosen by their opponent. They find a commitment device folk theorem: every individually rational (correlated) strategy in a basic two player game G can be obtained as a (Nash equilibrium) of an extended commitment game. In essence this equilibrium of the commitment device game takes the following form: in equilibrium every player is supposed to choose a particular commitment device that plays the equilibrium strategy if everybody did choose his ’assigned’ commitment device and minmaxes a player otherwise. This suggests that our results could be generalized beyond symmetric outcomes if we allow players’ preferences to depend on their position or if we allow for different player populations for different player positions. In addition we could allow for correlation devices and strategies that depend on these. In fact, if we allow directly for preferences over (potentially mixed or correlated) strategies the extension of our ”anything is stable”-theorem is straightforward. Consider the (non-expected utility) preference type, as allowed in von Widekind (2004), that has as a dominant strategy to play in a certain position whatever strategy the outcome specifies and minmaxes every other preference type. If the material payoff is individually rational this configuration is clearly stable. A further extension would be to consider a random distribution over games instead of a single one, where, however, players always know which game they are playing. Our argument hardly changes. Consider a metaoutcome (a combination of outcomes in all games) and a type that against his own type has preferences such that each of these outcomes constitutes a subjective equilibrium in the respective game, yet minimizes the expected material payoff of any other type in each game. This can be constructed very much as we construct our discriminating types in this paper. Again, this fully discriminating type stabilizes any meta-outcome with expected payoffs above the overall minmax payoff. Notice that in this scenario the material payoff of the incumbent can even be below the minmax value for some games, as long as the average payoff is greater than the expected payoff of a player who receives the minmax payoff in all games. 24
References Banerjee, A., and J. W. Weibull (1993): “Evolutionary selection with discriminating players,” Working Paper, Department of Economics, Harvard University, WP 1637. (2000): “Neutrally stable outcomes in cheap-talk coordination games,” Games and Economic Behavior, 32, 1–24. ¨ th (1998): “Is altruism evolutionary stable?,” Bester, H., and W. Gu Journal of Economic Behavior & Organization, 34, 193–209. Dekel, E., J. C. Ely, and O. Yilankaya (2007): “Evolution of Preferences,” Review of Economic Studies, forthcoming. Ely, J. C., and O. Yilankaya (2001): “Nash equilibrium and evolution of preferences,” Journal of Economic Theory, 97, 255–72. ¨ th, W. (1995): “An evolutionary approach to explaining cooperative Gu behavior by reciprocal incentives,” International Journal of Game Theory, 24, 323–44. ¨ th, W., and M. Yaari (1992): “Explaining reciprocal behavior in a Gu simple strategic game,” in Explaining Process and Change-Approaches to Evolutionary Economics, pp. 23–24. University of Michigan Press. Herold, F. (2003): “Carrot or Stick: The Evolution of Reciprocal Prefernces in a Haystack Model,” University of Munich Discussion Paper. Huck, S., and J. Oechsler (1999): “The Indirect Evolutionary Approach to Explaining Fair Allocations,” Games and Economic Behavior, 28, 13– 24. Kalai, A., E. Kalai, E. Lehrer, and D. Samet (2007): “A Commitment Folk Theorem,” Unpublished manuscript. Koc ¸ kesen, L., E. A. Ok, and R. Sethi (2000a): “Evolution of interdependent preferences in aggregative games,” Games and Economic Behavior, 31, 303–10. (2000b): “The strategic advantage of negatively interdependence preferences,” Journal of Economic Theory, 92, 274–99. Ok, E., and F. Vega-Redondo (2001): “On the evolution of individualistic preferences: An incomplete information scenario,” Journal of Economic Theory, 97, 231–54. 25
Robson, A. J. (1990): “Efficiency in evolutionary games: Darwin, Nash and the secret handshake,” Journal of Theoretical Biology, 144, 379–96. Sethi, R., and E. Somanathan (2001): “Preference evolution and reciprocity,” Journal of Economic Theory, 97, 273–97. von Widekind, S. (2004): “Evolution of non-expected utility preferences,” Bielefeld University IMW Working Paper #370.
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