Does moral play equilibrate?

. Some ﬁ nite and symmetric two-player games have no (pure or mixed) symmetric Nash equilibrium when played by partly morally motivated players. The reason is that the "right thing to do" may be not to randomize. We analyze this issue both under complete information between equally moral players and under incomplete information between arbitrarily moral players. We provide necessary and su ﬃ cient conditions for the existence of equilibrium and illustrate the results with examples and counter-examples.


Introduction
In economics and non-cooperative game theory, economic agents and players are usually assumed to be pure consequentialists, that is, to evaluate their alternative courses of action (consumption or production plans, strategies) exclusively in terms of the consequence for themselves and perhaps also for others. However, people may to some extent also be driven by deontological motivations, such as a wish to "do the right thing" in the given situation. Such a partly morally motivated participant in a public goods game may, for example, contribute the amount that would maximize the group's welfare if everybody would do likewise. An individual who acts accordingly, even when expecting others not to follow suit, is not necessarily irrational, or prey to "magical thinking". Such a person may simply have a goal function that gives some weight to Immanuel Kant's (1785) categorical imperative, to "act only on the maxim that you would at the same time will to be a universal law ". In standard public goods games such partly morally motivated individuals may be behaviorally indistinguishable from altruists, individuals who are pure consequentialists but who attach a positive value to other's well-being. However, in other interactions, a Kantian moralist may behave quite differently from an altruist. Take a 2 × 2 coordination game, where both players obtain payoff 1 if both use their first pure strategy, 2 if both use their second pure strategy, and otherwise zero. An altruist who expects the opponent to play the first pure strategy will do likewise. By contrast, a Kantian moralist may instead use the second pure strategy. This will result in material payoff zero to both, but the moralist may obtain psychological utility from behaving in a way he wishes all would in such interactions. If two stern moralists would play the coordination game, they would do just fine. However, in some games moralists of intermediate degree, known by both, may not even have a N a s he q u i l i b r i u m ,a n dt h i sm a ya l s ob et h ec a s ew h e np l a y e r ' sd e g r e eo fm o r a l i t yi s their private information.
We here explore exactly these questions, more precisely whether symmetric Nash equilibria exist in symmetric and finite games played by partly morally motivated players. As a formal representation of such players we use the Homo moralis preferences that Alger and Weibull (2013) showed are evolutionarily stable in populations under assortative random matching. 1 We establish the existence of symmetric Nash equilibria for certain game classes, when played by such players, and we also give examples of simple games with no such equilibria. Our main results, Theorems 1 and 2, establish necessary and sufficient conditions for the existence of symmetric Nash equilibrium between partly morally motivated players under incomplete information about others' degree of morality.

Definitions and preliminaries
In this note we consider finite and symmetric games. Let  = {1} be the set of pure strategies, and let ∆ be the associated unit simplex of mixed strategies, Here  = P  =1   ,w h e r e  is the :th unity (column) vector, and the superscript  denotes transpose. We write  for the zero vector (origin).
Let  be an  × -matrix with "material" payoffs, let  ∈ [0 1] be a player type, and consider the associated payoff function   : where  and  are(column)vectorsin∆.T h ep a r a m e t e r is the degree of morality of Homo moralis,w i t h =0representing pure self-interest, or Homo oeconomicus,a n d 1 The idea that moral values may have been formed by evolutionary forces can be traced back to at least Darwin (1871). More recent informal treatments include, to mention a few, Alexander (1987) and de Waal (2006).  =1representing pure (Kantian) morality, or Homo kantiensis (Alger and Weibull, 2013). Thus   ( ) is the payoff (or utility) to a player with degree of morality  when using strategy  against an opponent using strategy  in a symmetric game with (material) payoff matrix . Both player positions have the same set  of pure strategies,  is the matrix of material payoffs to the row player, and  =   of those to the column player.
For a given matrix  and degree of morality  ∈ [0 1],l e t  : ∆ ⇒ ∆ be the best-reply correspondence of Homo moralis of degree : Hence, a rational player with Homo moralis preferences of type  will use some strategy  in the subset   () if expecting the other player to use mixed strategy  ∈ ∆. By Weierstrass' maximum theorem,   () is a non-empty and compact set for every  ∈ [0 1] and  ∈ ∆. However, as will be seen shortly, this set is not always convex. We will study the existence and nature of fixed points under   ,thatispoin ts  ∈ ∆ such that  ∈   (). These are then the symmetric Nash equilibria when two Homines morales of the same degree of morality meet. By Berge's maximum theorem,   is upper hemi-continuous. For  =0the correspondence  0 is convex-valued. In fact, all its values are then sub-simplices, non-empty subsets of ∆ spanned by finitely many vertices. This is the standard setting of non-cooperative game theory, and as is well known, there exists at least one fixed point whenever  =0.

Games between equally moral players
The analysis in this section generalizes results for symmetric 2 × 2 gamesinSection4 of Alger and Weibull (2013). We here consider strategic interactions under complete information between two equally moral players who play a symmetric  ×  game in material payoffs, for any  ∈ N.W eb e g i nb ya2 × 2 example that illustrates that the correspondence   need not be convex-valued for positive degrees of morality.
Hence, this term is strictly convex in  and so is   ( ), for any given 0 and  ∈ ∆. Therefore, 1 and  2 are fixed points for 0 ≤  ≤  0 := min{ }, and there is only one fixed point for Since  =   is the payoff matrix of the column player, is welfare,d e fined as the sum of the two players' material payoffsw h e nb o t hu s e strategy  ∈ ∆.T h i sd e fines the welfare function  : ∆ → R. Accordingly, the payoff function of a Homo moralis with degree of morality  c a nb ew r i t t e ni nt h e form for every  ∈ ∆,s o existence of Nash equilibrium then follows immediately from Kakutani's fixed point theorem.

Proposition 1. The set of fixed points is non-empty and compact if  =0 .T h e same is true for every
We note beforehand that a sufficient condition for  to be concave is that the symmetric matrix  +   is negative-semidefinite. See Proposition 4 below for a more general result.
Hence, there exists at least one fixed point. From strict concavity of  we know that the sets   () are singletons for all  ∈ ∆ and 0.U s i n gfirst-order conditions, expressed in  1 only (with  2 =1−  1 ), we conclude Since  1 6 ∈ ∆ 1 and  2 6 ∈ ∆ 2 for all  ∈ [0 1], neither  1 nor  2 can be fixed points for any .A l lfixed points (and we know at least one exists) are thus found by solving  1 =  1 with  1 given by the necessary first-order condition (2). This leads to exactly one fixed point for every  ∈ (0 1],n a m e l y A game-theoretically important class of games in which  +   is negativesemidefinite are all constant-sum games (then  +   is the a matrix with identical entries), with zero-sum games as the most prominent special case.
Proposition 2. Let  be the payoff matrix of a symmetric constant-sum game. For any 1, the set of fixed points is identical with the non-empty set of fixed points when  =0, while every  ∈ ∆ is a fixed point when  =1.
In other words, all Homines morales, except Homo kantiensis, behave like Homo oeconomicus in all (finite and symmetric two-player) constant-sum games.
The remaining situation to investigate is thus when 0 and  is not concave. W ebe g i nw i t ha ne x a m p l e .

Example 3. Consider the generalized Rock-Scissors-Paper (RSP) game matrix
for any −1. We note that this is a constant-sum game if and only if  =0 . For  =0, the unique symmetric Nash equilibrium strategy is the barycenter   .A s is well-known, this unique equilibrium is unstable in the replicator dynamic for all 0 and asymptotically stable for all 0. 2 The function  is strictly concave if 0 a n ds t r i c t l yc o n v e xi f0,be c a u s ef o ra n y ∈ ∆: Proposition 1 ensures existence of at least one fixed point if the welfare function  is concave on ∆. If the welfare function instead is strictly convex, then fixed points may not exist. The next result provides necessary and sufficient conditions for existence in the latter case. The usefulness of both Propositions 1 and 3 depends on how easy or hard it is to verify that the welfare function is either concave or strictly convex on the unit simplex. Here are necessary and sufficient conditions for each of these properties.
Proof. First observe that for any 0 1 and any two { } ⊂ ∆,w eh a v e In some applications the payoff matrix  is symmetric;   = .I ns u c hpotential or partnership (or doubly symmetric)g a m e s ,i ti sw e l l -k n o w nt h a ta v e r a g ep a y o ff increases along all solution trajectories to the replicator dynamic (see e.g. Section 3.6 in Weibull, 1995). For such games and any positive degree of morality, any global welfare maximizer is a fixed point, and every fixed point is a local maximizer. Formally: implying    ≤   , and therefore   ( ) ≤   ( ) for all  ∈ ∆, i.e.,  ∈   ().
Next assume  ∈   ().T h e n is a global maximizer of   over ∆.I np a r t i c u l a r the directional derivative of   in the direction of −,ev aluatedat,isnotpositive, In case that ( − )   =0for some , also the second directional derivative of   in the direction of  − , evaluated at ,i sn o tp o s i t i v e , Now the two displayed inequalities are sufficient for  to be a local maximizer of  , as those inequalities are also statements about first and second directional derivatives of  .
In case of symmetric  there may indeed be fixed points  ∈   () that are local, but not global, maximizers of    subject to  ∈ ∆. This happens in Example 1 for small  ≥ 0.I f is not symmetric, neither (a) nor (b) needs to hold. Example 3 shows that (a) can be violated, and violation of both (a) and (b) for 0 ≤ 1 is demonstrated by Example 2 when  6 = . Alger and Weibull, 2013) establishes that at least one symmetric Nash equilibrium always exist between equally moral players.

Incomplete information about others' morality
We now consider strategic interactions between two Homines morales who only know their own degree of morality, not that of the opponent. We will call an individual's degree of morality the individual's type and use the canonical notation Θ =[ 0  1] for the type space. We endow Θ with its Euclidean topology and let  be a Borel probability measure on Θ, representing the type distribution in the population from which the players are drawn independently at random 3 .
A strategy is a Borel-measurable function  : Θ → ∆, assigning to each type  ∈ Θ a mixed strategy () ∈ ∆.ANash equilibrium under incomplete information is a strategy  that is a best reply to itself. A strategy  is optimal against a mixed strategy It follows from standard measurable-selection theory à la Kuratowski-Ryll-Nardzewski (see e.g. 18.3 and 18.4 in Aliprantis and Border, 2006, or 14.29 and 14.37 in Rockafellar and Wets, 2009) that such an optimal strategy  : Θ → ∆ exists for each  ∈ ∆.As t r a t e g y : Θ → ∆ is a best reply to itself, or constitutes a symmetric Nash equilibrium, if the following condition holds for all  ∈ Θ: By linearity of the payoff function with respect to , is the representative agent's mixed strategy. In other words, in order to be a best reply to itself, a strategy  : Θ → ∆ has to be optimal against its own representative agent's mixed strategy.
Existence is non-trivial. However one may characterize Nash equilibrium by way of first-and second-order optimality conditions. In order to state these, for each type  ∈ Θ let ()= · ( +   ), the Hessian matrix of   (·), for any  ∈ ∆. For any strategy  : Θ → ∆,l e t This is the gradient of the payoff   () with respect to  ∈ ∆, evaluated at  = (). For each pure strategy  ∈ ,l e t is a symmetric rank-two update of the Hessian  (),u s i n gt h e gradient () ∈ R  and the :th unit vector   ∈ ∆. Finally, for any strategy ,e a c h type  ∈ Θ and every pure strategy  ∈ ,w ed e fine the following polyhedral cone The result to follow establishes that, given any type distribution ,as t r a t e g y  : Θ → ∆ constitutes a Nash equilibrium under incomplete information if and only if three conditions are met: a first-order (Lagrangian) condition, a complementary slackness condition, and a second-order (curvature) condition. The reason why a second-order condition is sufficient is that all types' payoff functions are linearquadratic in their own strategy choice (in the underlying game). To ease reading, we split the result in two separate statements and provide a joint proof of them only after stating both.
Theorem 1. For any Borel probability measure  on ∆,as t r a t e g y : Θ → ∆ is a best reply to itself if and only if there are Borel-measurable functions  0 : Θ → R and   : Θ → R + for all  ∈  such that, for all pure strategies  and for all types : We say that a strategy  : Θ → ∆ is a better reply than  : Proposition 6. If (6) is violated for some pure strategy  and type ,t h e nt h e r e exists a better reply for this type , namely, the strategy  : Θ → ∆ that agrees with  for all types  6 =  but has Proof. The assertions in Theorem 1 follow from (Bomze 2016, Thm. 2.3), formulated for minimizing the negative −  (·) there; note that as ∆ is compact, we can ignore the index  =0dealing with unbounded feasible rays there. The case of ∆ has been dealt already in the previous papers (Bomze 1997a(Bomze ,1997b where also the arguments for Proposition 6 can be found.
We saw in Section 3 an example of a game that has no equilibrium for equally moral players, irrespective of their common degree of morality, as long as it is positive. In terms of the machinery in the present section, the observation can be recast as the statement that this game has no Nash equilibrium under incomplete information when the type distribution  places unit probability mass on some type 0.T h e following subsection deals with this case by providing conditions which simplify those of Theorem 1.

A homogeneous population.
Suppose now that  places unit probability on some  ∈ Θ. Applying the above general machinery, we search for a fixed point of the best-reply correspondence   , i.e. a strategy ∈ ∆ which coincides with both  (a) ( ) ∈ R +1 satisfies the linear equation system Finally, condition (c) is exactly condition (6) in Theorem 1.
Note that by construction   ()=  () ≤ 0 for all  ∈ Γ  () and all  ∈  as   ≥ 0 holds for all  ∈  \ . Hence condition (c) is ensured if ( +   ) is negative-semidefinite, as  ∈ Γ  () implies also   ≤ 0 and since On the other hand, already local optimality of for the function   () implies for 0 that [ +   ]  is negative-semidefinite on [  ] ⊥ , a linear subspace of codimension one in R  . This necessary condition can be viewed as a localized version (relative to the face of the simplex that contains in its relative interior) of the sufficient existence criterion in Proposition 1; see Proposition 4.

Remark 2.
In Example 3 we noted that no symmetric Nash equilibrium exists under complete information in a game between equally moral players when −1  0 and 0 1. Formally, such a situation can be represented as incomplete information with a Dirac measure placed on that particular type . Consider instead any continuous type distribution  on Θ =[0 1]. W em a yth end ivideth et ypespace into three disjoint intervals   with (  )=13,f o r =1 2 3.I fa l lt y p e si n  play pure strategy ,t h e na l lt y p e s ∈ Θ best respond to =   , the barycenter of the strategy simplex. Hence, the non-existence of symmetric equilibrium under complete information and equally moral players may be non-robust to arbitrarily small degrees of incomplete information about morality, as measured in the  1 -norm.