Games with Comparative Utility Functions

(1)

Helmut Meister

Games with Comparative Utility Functions

Lehrgebiet Stochastik Forschungsbericht

Fakultät für

Mathematik und

Informatik

(2)

Games with Comparative Utility Functions

Helmut Meister

Department of Mathematics and Computer Science FernUniversit¨at

Universit¨atsstraße 1 D-58084 Hagen, Germany email: meisterhelmut1@t-online.de

March 1, 2020

Abstract

The central objects of investigation of this paper are games with comparative utility functions. This concept can be considered as a relativistic approach to expected utility theory. The approach covers also the classical idea of utility functions. In some recent research papers the philosophic justification for games of this type has been discussed in more detail.

The notion of a Nash Equilibrium also applies to the context of non- cooperative games with comparative utility functions for the players. We will show that equilibria in the sense of Nash can be characterized by the solutions of complementarity problems in a very similar way as for classical non-cooperative games. Moreover, the question of existence of pure strategy Nash Equilibrium will be discussed. If the utility functions are realizations of random variables, the frequency of pure strategy equilibrium appearance for games with a large number of options follows the (1− ¹_e)-rule. This behavior has already been derived for classical non- cooperative games, but the stochastic properties are somewhat different for comparative games.

Keywords: Game Theory, Non-cooperative games, Nash Equilibrium Strate- gies, Pure Strategy Equilibrium, Preference Scheme, Comparative Expected Util- ity

1 Problem

Most approaches to the theory of preferences rely on the transitivity assumption, which grants that preferences of agents can be represented by utility functions and therefore opens the way to calculate expected utility values. As against this postulate of transitive preference schemes, experience of psychologists shows that in several situations preference schemes of agents are more complex (see

(3)

for instance [3], page 304 ff). The problem arises in the situation where agents build their preferences of actions according to their actual level of welfare or prefer to persist in their familiar behavior. In doing so they may end up in cycling preferences. This case is a bitch, because there exists no classical utility function which represents the preference scheme. Another criticism arises from Allais Paradoxon, which is also investigated in [3]. It concerns a choice problem designed by Maurice Allais (1953) and shows an inconsistency between actually observed choices of agents and the predictions of expected utility theory.

Therefore, it makes sense to study other approaches to represent preferences.

The proposal is to base preferences on comparative utility functions. The concept of comparative utility seems to be a more adequate approach to model situations with a status quo for players, from which they assess other options.

This approach makes use of a benchmark (or zero point of choiceworthiness) and the difference in utility for all other alternatives. The basic philosophy of this model has been discussed by D. Robert [10] (2018). A similar line of thought was already chosen in the author’s thesis paper [6] (1987). This concept can be considered as some kind of relativistic approach to expected utility theory. In the proposed model it may even appear that preferences on the space of options are not transitive and therefore do not result in overall best solutions. Neverthe- less, using randomized strategies, we can introduce a concept of optimality and show the existence of optimal strategies. In this sense the proposed approach follows the idea of expected comparative utility.

2 Existence of Optimal Strategies

We first introduce the concept of comparative utility.

2.1 Definition: We consider the case of a finite set S of strategies. A comparative utility functionis a function v:S×S→R.

That means, outcomes are ordered by relative utility values depending on the actual state rather than absolute utility values. It is not necessary fors∈S to assumev(s, s) = 0, because only the differences of utilities of all remaining options with respect to the given starting point s of the game are relevant for the agent. In this situation, the according preference scheme of an agent may not have a state which can be considered as most preferred for the agent.

Nevertheless, we can introduce a concept of expected comparative utility in a canonical way.

2.2 Definition: Let be given a comparative utility function v on a finite strategy set S :={1, . . . , n}. Then we can represent v by a matrix V with Vij :=

v(i, j), i, j ∈ S. This matrix will be called a comparative utility matrix.

Further, let∆_n be the unit simplex in the n-dimensional euclidean space. The expected comparative utility of p∈∆_n is then given by the vector pV, and thetotal expected comparative utilityis defined bypV p.

(4)

The interpretation of this notation constitutes as follows: pV is the expected relative utility vector given the randomized strategypof the agent, andpV pis his expected total return. In this sense we can additionally define an optimality condition for randomized strategies.

2.3 Definition: Let again be given a comparative utility matrix V to a finite strategy setS:={1, . . . , n}. Then, a randomized strategy p∈∆_n will be called abest strategyof the agent, if

pV p≥pV r ∀r∈∆n (1)

is satisfied.

Normally, pure optimal strategies will not always exist. However, with the aid of a standard argument frequently applied in game theory, the existence of optimal randomized strategies can easily be derived.

2.4 Theorem: For each comparative utility matrix V to a finite strategy set S:={1, . . . , n}, there exists a best strategy p∈∆n.

We will delay the proof to the game theoretic context discussed later. It makes use of Kakutani’s fixed point theorem, which draws on a compactness argument. Therefore, there exists no direct way to identify a best strategy, and we have to focus on other methods, if we want to calculate best strategies. It turns out that linear complementarity theory offers one of the most promising approaches.

2.5 Theorem: Let be given a comparative utility matrixV to a finite strategy setS:={1, . . . , n}. Ifp∈∆n is a best strategy withpV p >0, thenx:= _{pV p}^p is a solution to the linear complementarity problem

(LCP) xV = 1⁽ⁿ⁾−u x, u≥0,hx, ui= 0 (2) with the n-dimensional vector1⁽ⁿ⁾:= (1, . . . ,1).

On the other hand, if x≥0, x6= 0is a solution of (LCP), then p:= ^x h^x,1⁽ⁿ⁾i is a best strategy.

Proof. Letpbe a best strategy. Then

pV p≥pV b^(j) for all unit vectors b^(j)∈∆_n. (3) Moreover, we have

n

X

j=1

pV b^(j)pj =pV p. (4)

From (3) and (4), we conclude that

pj = 0 for pV b^(j)< pV p (5)

(5)

must be satisfied. Thereforex:=_{pV p}^p is a solution to (LCP).

On the other hand, let now x ≥ 0, x 6= 0 be a solution of (LCP). Then, settingp:= ^x

h^x,1⁽ⁿ⁾i, we get

pV = 1

x,1⁽ⁿ⁾1⁽ⁿ⁾−u with u≥0,hu, pi= 0 (6) as well as

pV p= 1

x,1⁽ⁿ⁾ (7) and

pV r= 1

x,1⁽ⁿ⁾− hu, ri ≤ 1

x,1⁽ⁿ⁾ =pV p ∀r∈∆_n, (8) which shows thatpis a best strategy.

The linear complementarity problem (LCP) offers a chance to compute all best strategies.

2.6 Remarks: 1. We solve the linear complementarity problem by a brute force method using a program developed in the NetLogo environment ([9]).

This approach is acceptable as long as the dimension of the matrix V is not too large. The program checks all n×n-sub-matrices of the matrix M :=

V I

built from rows ofV and the identity matrixI such that the indexes of the rows are complementary subsets S and S^c of {1, . . . , n}.

We denote the corresponding sub-matrix by VS

IS^c

. Then, all vectors (xS, uS^c) satisfying the linear equation

(xS, uS^c) VS

I_Sc

= 1⁽ⁿ⁾ (9)

with non-negative components represent a solution of (LCP). To this end, we have only to set all components ofx_Sc and those of u_S equal to zero and obtain the desired solution of (LCP).

2. It deserves to be mentioned that there exist some classical results con- cerning the number of solutions of linear complementarity problems. For more details, we refer to the publications of K. G. Murty [4], R. Saigal [11] and the PhD Thesis of the author [5].

We examine some special cases.¹

1In the following examples we address subsets of options by their indicator vectors.

(6)

2.7 Examples: We study different types of comparative utility functions representing the behavior of agents of varying character.

1. First, we illustrate the meaning of comparative utility by an often cited example, which models the situation of a population consisting of hawks and doves representing animals with rival behavior. This game may be considered as a persisting internal conflict between aggressive and defen- sive individuals about the ownership of some part of territory and the corresponding reproduction privilege. When playing this game we focus on the whole population as player rather than on the individual level.

This approach gives rise to formulate the game by a comparative utility function in a natural way. We denote aggressive behavior byAand defen- sive behavior by D. The corresponding reproduction probabilities after conflicts between different individuals of both types are assumed to be uAA, uAD, uDAanduDD. Hence, the comparative utility function for the whole population takes the form

V =

uAAuAD

uDAuDD

. (10)

The solution of the LCP in Theorem 2.5 (2) associated with the comparative utility functionV turns out to be the most profitable and therefore stable mix of individuals with behavior A and D for the population. If we assume that the conflict of twoA-types may result in serious injuries for both, the population survives better with moreD-types. On the other handA-types have a better standing againstD-types. This fact suggests us to assume uAA < uAD and uDA > uDD. In this case, the LCP can- not have a pure solution. Denoting the portion of A-types within the population byp, we find the solution of the LCP by the equation

p uAA+ (1−p)uDA=p uAD+ (1−p)uDD, (11) which leads to

p= uDA−uDD

uAD−uAA+uDA−uDD

. (12)

This result shows that the portion of A-types becomes small, whenever uAD is high in comparison to uAA, or in other words, whenever two A- types have high risks in conflicts.

2. The next example shows, that comparative utility functions cover also the classical case of absolute utility on the set of optionsS. We have only to set all rows of the utility matrixV to equal vectorsu∈R, resulting in

V =



 u ...

u



 (13)

(7)

The solution of the LCP in Theorem 2.5 (2) for u:=

1 2 3 4 turns out to be

Subset:

0 0 0 1

(14)

Solution vector x:

0 0 0 1 xV =

1 2 3 4 xV x= 4,

showing that the result is identical with the overall best option.

3. Next we consider the case of a conservative agent, who prefers to keep his option. This case can be modeled for instance by a utility matrix of the type

V =







5 2 3 4

1 5 3 4

1 2 5 4

1 2 3 5







(15)

and has a longer list of solutions

Subset LCP Solution x xV xVx

0 0 0 1

1 2 3 5

5

0 0 1 0

1 2 5 4

5

0 0 1 1

0 0 0.67 0.33

1 2 4.33 4.33 4.33

0 1 0 0

1 5 3 4

5

0 1 0 1

0 0.75 0 0.25

1 4.25 3 4.25 4.25

1 0 0 0

5 2 3 4

5

1 0 0 1

0.8 0 0 0.2

4.2 2 3 4.2 4.2

All pure strategies turn out to be solutions for the conservative agent, but also three mixed strategies are best strategies. In the extreme case when V is the n×n unit matrix all strategies with non-zero equal entries on some subset of{1, . . . , n} and zeroes elsewhere are best strategies.

4. In addition to the previous examples we examine the case of a completely volatile agent with a comparative utility matrix given by

V =







0 8 2 6 8

0 0 5 3 3

5 0 0 9 7

3 4 0 0 9

7 9 5 0 0







. (16)

The only best strategy in this case is a completely mixed strategy of the

(8)

kind

Subset:

1 1 1 1 1

(17) Solution vector:

0.06 0.36 0.23 0.04 0.32

xV =

3.49 3.49 3.49 3.49 3.49 xV x= 3.49

3 Comparative Games

The previously developed setting can be easily extended to the context of n- person games. The approach has the potential to set up a model of a game with comparative utility functions. Equilibrium conditions in the sense of Nash ([8]) can be formulated in the classical way.

3.1 Definition: Let be given a finite setN with |N| =:n of agents, each of them having a setSi:={1, . . . , ni}of options. For each agentiand for each n- tupelp⁽¹⁾, . . . , p⁽ⁿ⁾ of randomized strategies, we introduce the notation p⁽⁻ⁱ⁾:=

⊗_j∈N\{i}p^(j) for the product measure of the probability measures p^(j), j 6= i on the space S_−i :=Q

j∈N\{i}Sj. We suppose that each agent i ∈N has real coefficientsv_h,j,k⁽ⁱ⁾ representing his payoffs for pure strategyjagainstkdepending on the optionsh∈S_−i of all other agents. Then, theExpected Comparative Utility MatrixV⁽ⁱ⁾(p⁽⁻ⁱ⁾) is defined by its coefficients

V_j,k⁽ⁱ⁾(p⁽⁻ⁱ⁾) := X

h∈S−i

p⁽⁻ⁱ⁾(h)v⁽ⁱ⁾_h,j,k j, k= 1, . . . , ni. (18) This kind of game will be called aComparative Game. ANash Equilibrium of the game is an n-tupel of randomized strategiesp⁽¹⁾, . . . , p⁽ⁿ⁾ such thatp⁽ⁱ⁾ is a best strategy for the comparative utility matrixV⁽ⁱ⁾(p⁽⁻ⁱ⁾)for each agenti.

Comparative games have a specific payoff structure. This will be a matter of the following remark.

3.2 Remark: In a comparative game, each player i has a list of comparative utility matrices V_h⁽ⁱ⁾:=V⁽ⁱ⁾(b^(h)), where b^(h)=b^(j¹⁾⊗. . .⊗b^(jⁱ⁻¹⁾⊗b^(jⁱ⁺¹⁾⊗ . . .⊗b^(jⁿ⁾runs through all pure strategy combinations associated with options h= (j1, . . . , j_i−1, ji+1, . . . jn) from S_−i of his counter players. Therefore, given the probability measuresp^(j)(j ∈N), the expected comparative utility matrix V⁽ⁱ⁾(p⁽⁻ⁱ⁾) is the expectation of the matrices V_h⁽ⁱ⁾, h ∈ S_−i with respect to the probability measurep⁽⁻ⁱ⁾. Moreover, V⁽ⁱ⁾ is a multi-linear function of the argumentsp^(j), j6=i. The classical non-cooperative games withnplayers have a payoff structure different from comparative games. Particularly, the expected comparative utility matrices are replaced by simple expectation values in this case.

(9)

The following example illustrates the situation of comparative utility matrices.

3.3 Example: We model the situation of two neighboring countries each of which has to decide how the electric power supply will be organized in future.

Both have two alternatives, centralized nuclear power plants (C) or decentralized wind parks (D). If players have chosen one of the options, it is difficult to change the concept without facing additional investment. The comparative utilities for the model may take the following shape:

Payoffs for Player 1

Player 2 plays C Player 2 plays D

C D C D

C 90 50 C 90 95

D 80 100 D 50 100

and vice verse for player 2. In this model there exists some benefit, whenever a player switches to the option chosen by his neighbor, because there exists already some know-how to manage the concept. The game has several equilibria:

Player 1 Player 2 Payoff Player 1 Payoff Player 2

C C 90 90

C (0.33 0.66) 90 83.33

(0.33 0.66) C 83.33 90

D D 100 100

The solutions underpin the agreement of choices for both players. The mixed approach is not a very desirable equilibrium in both cases.

The existence theorem for Nash Equilibria has already been proven in [6]

in a more general context. Although, to keep the paper self contained, we will repeat the proof in our particular case.

3.4 Theorem: Each Comparative Game as specified in Definition 3.1 has at least one Nash Equilibrium.

Proof. For each agentiwe define the correspondenceFi:Q

i∈N∆n_i →2^∆ⁿⁱ by Fi(x⁽¹⁾, . . . , x⁽ⁿ⁾) :={q∈∆n_i|x⁽ⁱ⁾V⁽ⁱ⁾(x⁽⁻ⁱ⁾)q= max

r∈∆_nix⁽ⁱ⁾V⁽ⁱ⁾(x⁽⁻ⁱ⁾)r} (19)

∀(x⁽¹⁾, . . . , x⁽ⁿ⁾)∈∆n₁×. . .×∆n_n.

Then F1×. . .×Fn is an upper-semicontinuous, convex- and compact-valued correspondence and by Kakutani’s fixed point theorem ([7]) has therefore a point (p⁽¹⁾, . . . , p⁽ⁿ⁾)∈∆_n₁×. . .×∆_n_n with p⁽ⁱ⁾∈F_i(p⁽¹⁾, . . . , p⁽ⁿ⁾). Thus, we have shown that

p⁽ⁱ⁾V⁽ⁱ⁾(p⁽⁻ⁱ⁾)p⁽ⁱ⁾≥p⁽ⁱ⁾V⁽ⁱ⁾(p⁽⁻ⁱ⁾)r ∀r∈∆_n_i (20) holds.

(10)

This theorem applies directly to the particular case of Theorem 2.4 and provides the missing proof. A similar result as stated in Theorem 2.5 turns out to apply also in the more general case.

3.5 Theorem: Let be given a Comparative Game as given in Definition 3.1.

Then the following statements are valid:

1. Ifp⁽¹⁾∈∆₁, . . . , p⁽ⁿ⁾∈∆_nis a Nash Equilibrium withp⁽ⁱ⁾V⁽ⁱ⁾(p⁽⁻ⁱ⁾)p⁽ⁱ⁾>

0, i= 1, . . . , n, then the vectorsx⁽ⁱ⁾:= _p_(i)_V_(i)^p_(p⁽ⁱ⁾_(−i)_)p_(i) solve the complementarity problems

(N CP) x⁽ⁱ⁾V⁽ⁱ⁾(p⁽⁻ⁱ⁾) = 1⁽ⁿⁱ⁾−u⁽ⁱ⁾ (21) x⁽ⁱ⁾, u⁽ⁱ⁾≥0,D

x⁽ⁱ⁾, u⁽ⁱ⁾E

= 0 for i= 1, . . . , n.

2. On the other hand, if x⁽ⁱ⁾ ≥0, x⁽ⁱ⁾ 6= 0, p⁽ⁱ⁾ := ^x⁽ⁱ⁾

h^x⁽ⁱ⁾^,1⁽ⁿⁱ⁾i (i = 1, . . . , n) are solutions of (NCP), thenp⁽¹⁾, . . . , p⁽ⁿ⁾form a Nash Equilibrium of the Comparative Game.

The proof is a straight forward extension of that of Theorem 2.5.

We are now concerned with he question how Bimatrix Games can be inte- grated in the context of comparative games.

3.6 Example(Bimatrix Game): We start with a 2-player game, where player 1 hasmand player 2 hasnpure strategies, and the payoffs for player 1 is given by them×n-matrixA= (aij), the payoffs for player 2 are given by then×m- matrixB = (b_ji) accordingly. In this setting, player 1 is considered to be the row player and player 2 functions as column player for matrixAand vice verse forB. In the language of comparative games the coefficients of the comparative utilities for both players can be defined by

v⁽¹⁾_i,j,k:=akj (22)

v⁽²⁾_i,j,k:=bki

and are independent of the reference optionsiandj, respectively. Then, by the definition of comparative games, if player 2 chooses the strategy q ∈∆n, the expected comparative matrixV⁽¹⁾(q) of player 1 is given by

V_i,k⁽¹⁾(q) =

n

X

j=1

qjakj (23)

and thus takes the form

V⁽¹⁾(q) =



 qA^T

. . . qA^T



, (24)

(11)

with allm rows identical. The analogous argumentation for player 2 shows V_j,k⁽²⁾(p) =

m

X

i=1

pibki (25)

and

V⁽²⁾(p) =



 pB^T

. . . pB^T



, (26) whenever player 1 chooses strategy p ∈ ∆m. The expected payoffs for both players are therefore given by

pV⁽¹⁾(q)p^T =pAq^T (27)

and

qV⁽²⁾(p)q^T =qBp^T, (28)

respectively. Consequently, the equilibrium conditions for the comparative game reflect completely the classical concept of Nash Equilibrium for bimatrix games.

4 A Probabilistic Analysis of Pure Strategy Equi- librium

Pure strategy equilibrium has been a topic of analysis since the beginning of mathematical game theory. There exist many practical reasons to search for such equilibria, because randomized strategies are sometimes difficult to com- municate. Several investigations on pure strategy equilibria for classical non- cooperative n-person games with randomly generated payoffs have been performed. The results show that pure strategy Nash Equilibria appear generally rather frequently. In the publication of Dresher ([1]) the asymptotic probability of this event was specified as (1−¹_e), where e= 2.71828. . . is Euler’s number.

This means that for large strategy sets in round about 63% of all cases a pure best strategy exists. Other work focuses on large games (see for instance [13]).

Further results for games with a finite number of players and the frequency of the appearance of pure strategy equilibria are available for zero sum games with randomly generated payoffs (see [2] and [12]). In this case, pure strategy equilibria are relatively rare. Therefore, it makes sense to seek for similar probabilistic findings in the case of comparative utility matrices.

4.1 Remark: The most evident case is given by a single player with a comparative utility matrixV. As stated in Theorem 2.5, a pure best strategy exists, if and only if the corresponding LCP has a solution vectorxof the typex=αb⁽ⁱ⁾ with α > 0. This in turn is satisfied if and only if the matrix V has a row

(12)

Vi. such thatVii is the maximum of all row elements. Apart from the case of an absolute utility function as considered in Example 2.7(2), the existence of such a row is not generally ensured. But, if we consider the coefficients ofV to be independent identically distributed random variables in the unit interval [0,1], the frequency of a given rowVi. to have Vii as its maximum is given by

1

n, since the probability to be maximal is equal for all row elements. Hence, the probability for V to have no such row is (1− ¹_n)ⁿ and a total upper limit for this probability ise⁻¹ .

First, we make a note on the characterization of pure strategy equilibrium for comparative games, which is a straight forward consequence of Theorem 3.5.

4.2 Lemma: Let be given a comparative game as introduced in Definition 3.1.

Then, the strategy combinationb:= (b⁽¹⁾, . . . , b⁽ⁿ⁾)is a pure strategy equilibrium, if and only if for each playeri∈N the maximal coefficient of the rowb⁽ⁱ⁾W⁽ⁱ⁾ of the matrixW⁽ⁱ⁾:=V⁽ⁱ⁾(b⁽⁻ⁱ⁾)is given byb⁽ⁱ⁾W⁽ⁱ⁾(b⁽ⁱ⁾)^T.

This conclusion opens the way for a probabilistic analysis of pure strategy equilibria. To get a suitable setting for this analysis, we first switch to some stochastic concepts.

4.3 Definition: A probability distribution on[0,1]^m is called permutational invariant, if it is invariant under permutations of the standard base vectors in R^m.

Of course, the m-dimensional Lebesgue measure on [0,1]^mis permutational invariant. But there exists a wide class of other distributions inheriting this property. We derive a useful property of permutational invariant distributions.

4.4 Lemma: Let X := (X1, . . . , Xm)be a permutational invariant distributed random variable on a probability space (Ω,A, P)with values in[0,1]^m. Then it satisfies

P{X_i ≤X_s(i= 1, . . . , m)}=P{X_i≤X_t(i= 1, . . . , m)} (29) for arbitrary componentsXs andXt ofX.

Proof. Let be given a permutation π of the indices 1, . . . , m with π(s) = t.

Then, becausePX is permutational invariant, (Xπ(1), . . . , Xπ(m)) has the same distribution as (X1, . . . , Xm). Therefore,

P{Xi≤Xs(i= 1, . . . , m)}=P

X_π(i)≤X_π(s)(i= 1, . . . , m) = (30)

=P{Xi≤X_t(i= 1, . . . , m)}.

We will now apply these findings to comparative games with randomly generated utility matrices. We consider the game as given in Definition 3.1 and

(13)

examine the frequency of the appearance of an equilibrium with pure strategies for all players. First, we will establish a connection between the stochastic considerations above and comparative expected utility matrices.

4.5 Remark: Let be given a comparative game as in Definition 3.1. Further, let be given a playeriand a setb= (b⁽¹⁾, . . . , b⁽ⁿ⁾) of pure strategiesb^(j)∈∆_n_j of all playersj. If we start with permutational invariant randomly generated rows ofV⁽ⁱ⁾(b⁽⁻ⁱ⁾), we conclude from Lemma 4.4 that within the rowb⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾) of the matrixV⁽ⁱ⁾(b⁽⁻ⁱ⁾) the probability for each component to be maximal in the row is equal, and is therefore given by _n¹

i .

This insight will help to prove a general probabilistic result on existence of pure strategies within the set of equilibria.

4.6 Corollary: We consider again a comparative game as defined in 3.1. We assume that

1. the family of rowsb⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾)of the expected comparative utility matri- cesV⁽ⁱ⁾(b⁽⁻ⁱ⁾)varying over all pure strategy combinationsb= (b⁽¹⁾, . . . , b⁽ⁿ⁾) andi∈N is a family of independent random variables,

2. for all given pure strategy combinationb= (b⁽¹⁾, . . . , b⁽ⁿ⁾)the rowsb⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾) withi∈N are permutational invariant random variables with values from [0,1]ⁿⁱ.

Then, the probability that the game has a Nash Equilibrium with pure strategies for all players is given by

p:= 1−

1− 1

Q

i∈Nni

^Qi∈Nn_i

. (31)

Proof. Let be given a strategy combination b = (b⁽¹⁾, . . . , b⁽ⁿ⁾) of all players consisting of pure strategies. We will derive the probability that b is an equilibrium of the game. Since, by assumption2., the rows b⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾), i ∈ N are permutational invariant distributed random variables with values in [0,1]ⁿⁱ, we conclude from Remark 4.5 that the probability of any component of the row b⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾) to be maximal in this row is equal to _n¹

i. In order b = (b⁽¹⁾, . . . , b⁽ⁿ⁾) to build a pure strategy equilibrium, it is necessary and sufficient by Lemma 4.2 that the i-th component b⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾)(b⁽ⁱ⁾)^T is the maximal component ofb⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾).

Now, since by assumption 1., the family b⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾)(i ∈ N) consists of independent random variables, the probability for b⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾)(b⁽ⁱ⁾)^T to be maximal in the rowb⁽ⁱ⁾V⁽ⁱ⁾(b⁽⁻ⁱ⁾) for eachi ∈N is given by Q ¹

i∈Nni. Hence, the probability that the given strategy combination b does not satisfy all the maximum conditions for all players i ∈ N, can be calculated by 1−Q ¹

i∈Nni. So, again from the independence assumption 1., we get, that the probability

(14)

for failure of all possible strategy combinations b = (b⁽¹⁾, . . . , b⁽ⁿ⁾) with pure strategiesb⁽ⁱ⁾, i∈N is equal to

1−Q ¹

i∈Nni

^Qi∈Nn_i

.

It deserves to be mentioned, that the random variables as required in Corol- lary 4.6 can easily be constructed. There is no specific requirement for the coefficients of the random matrices to be uniformly distributed. Therefore, the approach also covers the case of the use of discrete random variables for matrix generation. Moreover, the result has some practical consequences. It shows that the probability for existence of at least one pure strategy equilibrium in comparative games is always greater than 1−e⁻¹≈63% irrespective of the number of pure strategies of all players. Therefore, we get some justification to search for pure strategy equilibria running through all pure strategy combinations of all players and verifying the conditions given in Lemma 4.2. The amount of calculations is of polynomial complexity depending on the number of all pure strategies for all players.

The result does not apply to classical non-cooperative n-person games, and particularly bimatrix games, since the independence assumption of Corollary 4.6 of the rows of these matrices is not satisfied, as Example 3.6 shows. In this sense, the stochastic behavior for the situation of comparative games is not a contradiction to the investigations of Dresher ([1]). But both results show, that the probabilities of pure strategy equilibrium existence converge to the same limit 1−e⁻¹for large games. Example 3.3 shows that a mixed strategy of the second player may also have a pure response by the other player. A probabilistic analysis of the occurrence of such equilibria has not yet be performed, but seems to be possible by a similar approach.

5 Concluding Remarks

We have shown that games with comparative utility functions for players can be considered as an extension of the classical notion of non-cooperative games with absolute utility functions. Nash equilibria can be characterized by solutions of complementarity problems similar to the classical case. The difference between both types emerges from the greater complexity of the utility functions.

Particularly, the Nash Equilibria of bimatrix games can be identified solving a corresponding linear complementarity problem. This is no longer valid for comparative two-person games. Only a single player can find his best strategies in a comparative game via a linear complementarity problem. Nevertheless, the asymptotic behavior of the frequency of pure strategy Nash Equilibrium appearance coincides with that of classical non-cooperative games.

(15)

References

[1] Dresher, M., 1970, Probability of a Pure Equilibrium Point in n-Person Games, Journal of Combinatorial Theory 8, 134-145.

[2] Hofri, M., 2006,On the Distribution of a Saddle Point Value in a Random Matrix, Department of Computer Science, WPI 100 Institute Road, Worces- ter MA 01609-2280, e-mail:hofri@cs.wpi.edu.

[3] Lewis, M., Vogel, S. (Translator), 2018,Rebellen des Denkens: Wie Daniel Kahneman und Amos Tversky die Psychologie revolutionierten, Goldmann Taschenbuch.

[4] Murty, K. G., 1972, On the Number of Solutions to the Complementarity Problem and Spanning Properties of Complementary Cones, Linear Algebra and its Applications 5, 65-108

[5] Meister, H., 1984,Zur Theorie des parametrischen Komplementarit¨atsprob- lems, Math. Systems in Economics 85

[6] Meister, H., 1987,The Purification Problem for Constrained Games with In- complete Information, Lecture Notes in Economics and Mathematical Sys- tems 295.

[7] Kakutani, Shizuo, 1941,A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal. 8 (3), 457–459, doi:10.1215/S0012-7094-41- 00838-4.

[8] Nash, John Forbes, 1950, Non-cooperative games, Dissertation, Princeton University.

[9] Wilensky, U., 1999, NetLogo, http://ccl.northwestern.edu/netlogo/, Center for Connected Learning and Computer-Based Modeling, Northwestern Uni- versity, Evanston, IL.

[10] Robert, D., 2018,Expected Comparative Utility Theory: A New Theory of Rational Choice, The Philosophical Forum published by Wiley Periodicals, Inc. on behalf of Philosophical Forum Inc (PF).

[11] Saigal, R.,A Characterization of the Constant Parity Property of the Num- ber of Solutions to the Linear Complementarity Problem, SIAM Journal 23, 40 - 45.

[12] Thorp, E. O., 1979, The probability that a matrix has a saddle point, Information Sciences Volume 19, Issue 2, Pages 91-95, https://doi.org/10.1016/0020-0255(79)90006-9.

[13] Villegas, A. J. R., Torres-Mart´ınez, J. P., 2010,A Direct Proof of the Exis- tence of Pure Strategy Equilibria in Large Generalized Games with Atomic Players, Working Papers wp311, University of Chile, Department of Eco- nomics.