(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations
aa r X i v : . [ c s . G T ] S e p (In)Existence of Equilibria for 2-Players, 2-Values Games withConcave Valuations ∗ Chryssis Georgiou † Marios Mavronicolas ‡ Burkhard Monien § September 10, 2020
Abstract
We consider , minimization games where the players’ costs take on twovalues, a, b , a < b . The players play mixed strategies and their costs are evaluated by unimodalvaluations . This broad class of valuations includes all concave, one-parameter functions F :[0 , → R with a unique maximum point. Our main result is an impossibility result statingthat: • If the maximum is obtained in (0 ,
1) and F (cid:0) (cid:1) = b , then there exists a 2-players, 2-valuesgame without F -equilibrium.The counterexample game used for the impossibility result belongs to a new class of very sparse2-players, 2-values bimatrix games which we call normal games .In an attempt to investigate the remaining case F (cid:0) (cid:1) = b , we show that: • Every normal, n -strategies game has an F -equilibrium when F (cid:0) (cid:1) = b . We present a lineartime algorithm for computing such an equilibrium. • For 2-players, 2-values games with 3 strategies we have that if F (cid:0) (cid:1) ≤ b , then every 2-players, 2-values, 3-strategies game has an F -equilibrium; if F (cid:0) (cid:1) > b , then there exists anormal 2-players, 2-values, 3-strategies game without F -equilibrium.To the best of our knowledge, this work is the first to provide an (almost complete) answer onwhether there is, for a given concave function F , a counterexample game without F -equilibrium. Concave valuations are concave functions from probability distributions to reals. In general, concavefunctions find many applications in Science and Engineering. For example, in MicroeconomicTheory, production functions are usually assumed to be concave, resulting in diminishing returnsto input factors; see, e.g., [14, pp. 363–364]. Also, their concavity makes concave valuations ∗ This work is partially supported by the German Research Foundation (DFG) within the Collaborative ResearchCentre “On-the-Fly-Computing” (SFB 901), and by funds for the promotion of research at the University of Cyprus. † Dept. of Computer Science, University of Cyprus, Nicosia, Cyprus; [email protected] ‡ Dept. of Computer Science, University of Cyprus, Nicosia, Cyprus; [email protected] § University of Paderborn, Paderborn, Germany; [email protected] risk – see, e.g., [8, 10, 17]. Furthermore, concave functions are utilizedin transportation networks (see, e.g., [16]) and in wireless networks (see, e.g., [3]).In this work, we study -players, -values minimization games with concave valuations for theirexistence of equilibria. Equilibria are special states of the system where no entity has an incentiveto switch. A (minimization) game may or may not have an equilibrium under a concave valuation(cf., [6, 12, 13]).For our investigation on the conditions for existence of equilibria we consider 2 -values games with unimodal valuations : • -values games are games where the players costs take on two values a and b , with a < b .We consider a subclass of counterexample 2-values games, termed normal games , where ( i )there is no ( a, a ) entry in the cost bimatrix, ( ii ) player 1 (the row player) has exactly one a entry per column, and ( iii ) player 2 (the column player) has exactly one a entry per row. So,normal games are very sparse 2-values games offering a large degree of simplicity, which weuse as a tool for deriving our main impossibility result. • Unimodal valuations are valuations that can be expressed as a one-parameter concave function F with a unique maximum point. Many valuations, like E + γ · Var and E + γ · SD , aredemonstrated to be unimodal; E , Var , and SD denote expectation, variance , and standarddeviation, respectively, and γ > risk factor .The question about the existence of equilibria was first put forward by Crawford [6], who gave anexplicit counterexample game without an equilibrium for some concave valuation. Crawford’s gamewas used as a gadget in N P -hardness proofs of the equilibrium existence problem for the explicit valuations E + γ · Var and E + γ · SD [12] and CVaR α [13]; CVaR α denotes conditional valuation-atrisk and α ∈ (0 ,
1) is the confidence level . (It should be noted that the given reductions in [12, 13]yield non-sparse games.) In contrast, our work deals with an abstract unimodal valuation V ; this isthe first investigation into games without an equilibrium under a general class of concave functions.Previous work has addressed specific concave valuations such as E + γ · Var [11, 12] and
CVaR α [13].Our results provide an almost complete answer to the equilibrium existence problem for 2-players, 2-values games with concave valuations. In short, we define a sequence of normal games,and show that for every unimodal valuation F with F (cid:0) (cid:1) = b , there exists a game from thesequence that does not have an F -equilibrium (Section 4). For the remaining case F (cid:0) (cid:1) = b , weprovide matching existence results (Section 5). We proceed with a detailed exposition of our results. Given a unimodal valuation V expressed bythe concave function F , we consider the value of x , denoted by x ( F ), where F takes the maximumvalue in [0 , x ( F ) on the existence of V -equilibria or, as wecall them, F -equilibria. We obtain the following impossibility result as our main result: • If x ( F ) > F (cid:16) (cid:17) = b , then there is a normal 2-players, 2-values game without F -equilibrium (Theorem 4.4).The impossibility result is shown using a family { C m : m ∈ N } of ( m + 1)-strategies games; C m is defined as the 1-row, 1-column extension of a Toepliz combinatorial bimatrix, D m , which defines2n m -strategies normal game. Interestingly, the bimatirx game defined by D m has a uniform F -equilibrium, while in Theorem 4.1 we prove necessary and sufficient conditions for the uniquenessof this uniform equilibrium. The impossibility result follows by proving, in Theorem 4.3, necessary and sufficient conditions guaranteeing that C m has no F -equilibrium. For example, C = ( a, b ) ( b, a ) ( a, b )( b, a ) ( a, b ) ( b, b )( b, b ) ( b, b ) ( b, a ) has no F -equilibrium if and only if F (cid:18) (cid:19) > b. To the best of our knowledge these are the first examples of 2-values games without equilibriumfor a class of concave valuations. The Crawford game has 3 values.We compliment our main result by observing that if x ( F ) = 0, then an F -equilibrium exists for all2-players, 2-values games and is PPAD -hard to compute (Theorem 3.8).Our main result leaves open the remaining case F (cid:0) (cid:1) = b , for which we obtain matching existenceresults: • Every 2-players, 2-values, n -strategies normal game has an F -equilibrium when the unimodalvaluation F satisfies F (cid:0) (cid:1) = b , and it can be computed in O ( n ) time (Corollary 5.8). Thisexistential result is derived in Theorem 5.7, where we determine efficiently a winning pair ,which defines an F -equilibrium. The algorithm runs in time O ( n ) and the supports in thewinning pair we determine have size 2. • Every 2-players, 2-values, 3-strategies game has an F -equilibrium when the unimodal valuation F satisfies F (cid:0) (cid:1) ≤ b (Theorem 5.9). The proof of this result is purely combinatorial. Thisresult nicely illuminates Theorem 4.3, which states that for F (cid:0) (cid:1) > b , the game C , having 3strategies, has no F -equilibrium. Win-lose games are the special case of 2-values games where a = 0 and b = 1. Win-lose gamesare as powerful as 2-values games when E -equilibria are considered. Abbott et al [1] show thatcomputing an E -equilibrium for a given win-lose bimatrix game is PPAD -complete.Two different models of sparse win-lose bimatrix games have been considered in [4, 5]. In themodel of Chen et al. [4], for the row player each row, and for the column player each column of thepayoff matrix has at most two 1’s entries. In the model of Codenotti et al. [5], there are at mosttwo 1’s per row and per column for both players. So the model in [5] is inferior to the model in [4].Recall that in our model, there is at most one a per column for the row player, and at most one a per row for the column player. So our model is not comparable with the models in [4, 5]. In bothpapers [4] and [5], polynomial time algorithms are given for win-lose games from the correspondingmodel of sparseness.Chen et al. [4] also show that approximating a Nash equilibrium is PPAD -hard in the modelwhere both matricies contain at most 10 non-zero entries in each row and in each column. Liu andSheng [9] extend this result to win-lose games when both matrices contain O (log n ) 1’s entries ineach row and in each column. Bil`o and Mavronicolas [2] prove that is N P -complete to decide if asimultaneously win-lose and symmetric 2-players game has an E -equilibrium with certain properties.3 .4 Paper Organization Section 2 presents the game-theoretic framework, together with some simple properties regardingequilibria for 2-players, 2-values games, and the definition of normal games. In Section 3 we defineunimodal valuations, we demonstrate that common risk-averse valuations are unimodal valuations,prove some simple properties of unimodal valuations, and consider the case of x ( F ) = 0. Our mainimpossibility result is presented in Section 4. Section 5 presents matching existence results. Weconclude in Section 6. For an integer ν ≥
2, an ν -players game G , or game, consists of (i) ν finite sets { S k } k ∈ [ ν ] of strategies, and (ii) ν cost functions { µ k } k ∈ [ ν ] , each mapping S = × k ∈ [ ν ] S k to the reals. So, µ k ( s ) isthe cost of player k ∈ [ ν ] on the profile s = h s , . . . , s ν i of strategies, one per player. All costs arenon-negative. Assume, without loss of generality, that for each player k ∈ [ ν ], S k = { , . . . , n − } ,with n ≥ mixed strategy for player k ∈ [ ν ] is a probability distribution p k on S k ; the support of player k in p k is the set σ ( p k ) = { s k ∈ S k | p k ( s k ) > } . Denote as p s k k the pure strategy of player k choosing the strategy s k with probability 1. A mixed profile is a tuple p = h p , . . . , p ν i of ν mixedstrategies, one per player. The mixed profile p induces probabilities p ( s ) for each profile s ∈ S with p ( s ) = Q k ∈ [ ν ] p k ( s k ). For a player k ∈ [ ν ], the partial profile s − k (resp., partial mixed profile p − k ) results by eliminating the strategy s k (resp., the mixed strategy p k ) from s (resp., from p ).For a player k ∈ [ ν ], a valuation function, or valuation for short, V k is a real-valued function,yielding a value V k ( p ) to each mixed profile p , so that in the special case when p is a profile s , V k ( s ) = µ k ( s ). The expectation valuation is defined as E k ( p ) = P s ∈ S p ( s ) µ k ( s ), with k ∈ [ ν ]. A valuation V = h V , . . . , V ν i is a tuple of valuations, one per player; G V denotes G together with V .We shall view each valuation V k , with k ∈ [ ν ], as a function of the mixed strategy p k , for a fixedpartial mixed profile p − k . The valuation V is concave if the following condition holds for everygame G : For each player k ∈ [ ν ], the valuation V k ( p k , p − k ) is concave in p k , for a fixed p − k .A V k -best response is a (pure or mixed) strategy p k that minimizes V k ( p k , p − k ) for a fixed p k . In this work, we shall consider minimization games, where each player k ∈ [ ν ] is interested inminimizing V k ( p ), that is, each player seeks to minimize her cost.The mixed profile p is a V -equilibrium if for each player k ∈ [ ν ], the mixed strategy p k is a V k -best response to p − k ; so, no player could unilaterally deviate to another mixed strategy toreduce her cost. In this case, we will say that player k is V -happy with p . We will be referring to an E -equilibrium when the valuation is the expectation valuation (and E -happy is defined analogously).A pure equilibrium is an equilibrium where all players play pure strategies.We shall consider two properties defined in reference to best-responses: Definition 2.1 (The Weak-Equilibrium-for-Expectation Property [11, Section 2.6])
The valuation V has the Weak-Equilibrium-for-Expectation property if the following condition Recall that a function f : D → R on a convex set D ⊆ R m is concave if for every pair of points x, y ∈ D , for all λ ∈ [0 , , f ( λy + (1 − λ ) x ) ≥ λ f ( y ) + (1 − λ ) f ( x ). olds for every game G : For each player k ∈ [ ν ] and a partial mixed profile p − k , if p k is a V k -bestresponse to p − k , then, for each pair of strategies ℓ, ℓ ′ ∈ σ ( p k ) , E k ( p ℓk , p − k ) = E k ( p ℓ ′ k , p − k ) . We shall often abbreviate the
Weak-Equilibrium-for-Expectation property as
WEEP . We shall omitreference to the game G when immaterial. Definition 2.2 (The Optimal-Value Property [12, Section 3.2])
The valuation V has theOptimal-Value property if the following condition holds for every game G : For each player k ∈ [ ν ] ,and a partial mixed profile p − k , if b p k is a V k -best response to p − k , then, for any mixed strategy q k with σ ( q k ) ⊆ σ ( b p k ) , V k ( q k , p − k ) = V k ( b p k , p − k ) . We recall from [12, Proposition 3.1]:
Proposition 2.1 (Concavity ⇒ Optimal-Value)
Assume that the valuation V k ( p k , p − k ) isconcave in p k . Then, V k has the Optimal-Value property. In this work we focus on 2-players games with ν = 2, where for each player k ∈ [2], k denotesthe other player. A 2-players game G will be denoted as ( α ij , β ij ) ≤ i,j Assume that the valuation V has the WEEP property. Then, for each player k ∈ [2] and a partial mixed profile p − k , if p k is a V k -best-response to p − k , then no strategy in σ ( p k ) dominates some strategy in σ ( p k ) with respect to σ ( p k ) . We now prove two simple properties used later in the paper. We begin with a sufficient conditionfor a pure equilibrium for a 2-players, 2-values game: Lemma 2.3 Consider a -players, -values game. If there is an E -equilibrium h p , p i with | σ ( p ) | = 1 or | σ ( p ) | = 1 , then there is also a pure equilibrium. Proof. Without loss of generality, assume that | σ ( p ) | = 1 with σ = { } . By the WEEP ,there is α ∈ { a, b } such that µ (0 , j ) = α for all j ∈ σ ( p ). Since p is an E -equilibrium, thisimplies that α ≤ µ (0 , j ) for all strategies j ∈ { , . . . , n − } . Take now b j ∈ σ ( p ) so that µ (0 , b j ) = min j ∈ σ µ (0 , j ). We prove that the profile (0 , b j ) is a pure equilibrium:5 Player 2 is happy with (0 , b j ): We have to prove that µ (0 , b j ) ≤ µ (0 , j ) for all strategies j ∈ { , . . . , n − } . Since b j ∈ σ and p is an E -equilibrium, µ (0 , b j ) = E ( p , p b j ) ≤ E ( p , p j ) = µ (0 , j ). • Player 1 is happy with (1 , b j ): We have to prove that µ (0 , b j ) ≤ µ ( i, b j ) for all strategies i ∈ { , . . . , n − } . This is vacuous if µ (0 , b j ) = a . So assume that µ (0 , b j ) = b . By thedefinition of b j , it follows that µ (0 , j ) = b for all j ∈ σ ( p ). Hence, E ( p , p ) = b . Since p is an E -best response to p , it follows that E ( p i , p ) ≥ b for all strategies i ∈ { , . . . , n − } .This implies that µ ( i, j ) = b for all pairs of strategies i ∈ { , . . . , n − } and j ∈ σ ( p ).Since b j ∈ σ ( p ), this implies that µ ( i, b j ) = b for all strategies i ∈ { , . . . , n − } . Hence, µ (0 , b j ) = µ ( i, b j ) for all strategies i ∈ { , . . . , n − } .The claim follows.We continue to prove a simple property of E -equilibria: Lemma 2.4 Consider a -players, -values game, with an E -equilibrium p = h p , p i with | σ ( p ) | =2 . Define the mixed strategy e p for player with σ ( e p ) = σ ( p ) and e p ( j ) = 12 for all j ∈ σ ( e p ) .Then, h p , e p i is also an E -equilibrium. Proof. Without loss of generality, take σ ( p ) = { , } . To prove that h p , e p i is an E -equilibrium,We consider each player separately: • Player 2 is E -happy: By the WEEP for ( p , p ), E ( p , p ) = E ( p , p ); since h p , p i is an E -equilibrium, E ( p , p ) ≤ E ( p , p ℓ ) for all strategies ℓ σ ( p ). Since σ ( p ) = σ ( e p ), theclaim follows. • Player 1 is E -happy: By the WEEP , there is a number α ∈ R with E ( p i , p ) = α for allstrategies i ∈ σ ( p ). By the linearity of Expectation, there are only three possible cases: (1) µ ( i, 0) = µ ( i, 1) = a for all i ∈ σ ( p ). (2) µ ( i, 0) = µ ( i, 1) = b for all i ∈ σ ( p ). (3) µ ( i, = µ ( i, 1) for all i ∈ σ ( p ).It follows that E (cid:0) p i , e p (cid:1) = a if µ ( i, 0) = µ ( i, 1) = a for all i ∈ σ ( p ) b if µ ( i, 0) = µ ( i, 1) = b for all i ∈ σ ( p ) a + b µ ( i, = µ ( i, 1) for all i ∈ σ ( p ) . So in all Cases (1) , (2) and (3) , E ( p i , e p ) is constant over all strategies i ∈ σ ( p ), and the WEEP holds, so that E ( p , e p ) = E ( p i , e p ) for any strategy i ∈ σ ( p ). Thus, it remains toprove that player 1 cannot improve by switching to some strategy ℓ σ ( p ). Clearly, E (cid:16) p ℓ , e p (cid:17) = a if µ ( ℓ, 0) = µ ( ℓ, 1) = ab if µ ( ℓ, 0) = µ ( ℓ, 1) = ba + b µ ( ℓ, = µ ( ℓ, . (1) . Note that in Case (2) , it must also hold that µ ( ℓ, 0) = µ ( ℓ, 1) = b for all strategies ℓ σ ( p ) since h p , p i is an E -equilibrium. So player 1 cannot improve inthis case either. In Case (3) , since h p , p i is an E -equilibrium, it must hold that for allstrategies ℓ σ ( p ), either µ ( ℓ, 0) = µ ( ℓ, 1) = b or µ ( ℓ, = µ ( ℓ, We now introduce a restriction of 2-players, 2-values games to very sparse games that we call normal games . Definition 2.3 (Normal Game) A 2-players, 2-values game G is normal if it fulfills: (1) There is no ( a, a ) entry in the bimatrix of G . (2) Player (the row player) has exactly one a entry per column. (3) Player (the column player) has exactly one a entry per row. Note that the definition of a normal game is symmetric with respect to the two players. Also notethat excluding ( a, a ) entries, excludes the trivial existence of pure equilibria. We prove: Lemma 2.5 Consider the normal game G V , where V has the WEEP . Then, G V has no V -equilibrium h p , p i with | σ ( p ) | = 1 or with | σ ( p ) | = 1 . Proof. Assume, by way of contradiction, that G has a V -equilibrium h p , p i with | σ ( p ) | = 1. (Bythe symmetry in the definition of a normal game, this assumption is with no loss of generality.)Let σ ( p ) = { i } . We proceed by case analysis. Assume first that | σ ( p ) | = 1. Let σ ( p ) = { j } .By Condition (1) for a normal game, either µ ( i, j ) = b or µ ( i, j ) = b . If µ ( i, j ) = b , then, byCondition (2) for a normal game, there is a strategy b i with µ ( b i, j ) = a . Hence, player 1 improvesby switching to strategy b i . A contradiction. The case where µ ( i, j ) = b is handled identically,using Condition (3) for a normal game.Assume now that | σ ( p ) | > 1. By the WEEP , there is an α ∈ { a, b } with µ ( i, j ) = α for all j ∈ σ ( p ). Since | σ ( p ) | > 1, Condition (3) for a normal game implies that α = b . Again byCondition (3) for a normal game, there is a strategy b j ∈ σ ( p ) with µ ( i, b j ) = a . Hence, player 2improves by switching to strategy b j . A contradiction.Note that Lemma 2.5 implies that a normal game has no pure equilibrium and that | σ ( p k ) | ≥ , k ∈ [2], for any V -equilibrium h p , p i . 7 Unimodal Valuations For each player k ∈ [2], define x k ( p , p ) := X ( s ,s ) ∈ S × S | µ k ( s ,s )= a p ( s ) · p ( s ) . Note that x k is linear in each of its arguments: for each λ ∈ [0 , x k ( λ · p ′ k + (1 − λ ) · p ′′ k , p ¯ k ) = λ · x k ( p ′ k , p ¯ k ) + (1 − λ ) · x k ( p ′′ k , p ¯ k ) . We proceed to define: Definition 3.1 (One-Parameter Valuation) For a -players, -values game, a valua-tion V is one-parameter if for each mixed profile h p , p i , for each player k ∈ [2] , V k ( p , p ) can be written as V k ( p , p ) = F ( x k ( p , p )) , for some function F : [0 , → R , with F (0) = b and F (1) = a . We observe: Lemma 3.1 For a -players, -values game, a one-parameter valuation V is concave if and onlyif F is concave. Proof. Consider the mixed profile h p k , p k i := ( λ · p ′ k + (1 − λ ) p ′′ k , p k ), with λ ∈ [0 , x k in p k , for each player k ∈ [2]. Since V is a one-parameter valuation, for k = 1, weget that F ( x ( p , p )) ≥ λ · F ( x ( p ′ , p )) + (1 − λ ) · F ( x ( p ′′ , p ))if and only if V ( p , p ) ≥ λ · V ( p ′ , p ) + (1 − λ ) · V ( p ′′ , p ) . Hence, V is concave if and only if F is concave. (The case for k = 2 is symmetric.)We now introduce a restriction of one-parameter valuations. Definition 3.2 (Unimodal Valuation) In the 2-values case, a one-parameter valuation V is unimodal if F : [0 , → R is a concave function with a unique maximum point. By Lemma 3.1, a unimodal valuation V is concave. We shall often identify the unimodal valuation V with the function F and refer to a V -equilibrium for a unimodal valuation V as an F -equilibrium. Finally, we define x = x ( F ) to be the value of x where F ( x ) takes its maximal value in [0 , .2 Examples Recall the Expectation E , Variance Var and Standard Deviation SD valuations. We shall considerthe valuations EVar γ = E + γ · Var and ESD γ = E + γ · SD , with γ > 0. Both EVar γ and ESD γ are concave functions as the sums of two concave functions; hence, they have the WEEP and the Optimal-Value property (Proposition 2.1). We first derive formulas for E , Var and SD in terms of x . Fix a mixed profile h p , p i . Then, E ( p , p ) = X ( s ,s ) ∈ S × S p ( s ) · p ( s ) · µ ( s , s )= a · X ( s ,s ) ∈ S × S | µ ( s ,s )= a p ( s ) · p ( s )+ b · X ( s ,s ) ∈ S × S | µ ( s ,s )= b p ( s ) · p ( s )= ( a − b ) · x ( p , p ) + b . So E is a one-parameter valuation with F ( x ) = ( a − b ) · x + b . Note that F ( x ) is strictly monotonedecreasing in x for x ∈ [0 , Var ( x ) = E ( x ) − ( E ( x )) for a random variable x , to derive Var ( p , p ) = ( a − b ) · x ( p , p ) + b − (( a − b ) x ( p , p ) + b ) = ( a − b ) · x ( p , p ) + b − ( a − b ) x ( p , p ) − a − b ) · b · x ( p , p ) − b = x ( p , p )(1 − x ( p , p ))( b − a ) , which implies SD ( p , p ) := p Var ( p , p ) = ( b − a ) p x ( p , p )(1 − x ( p , p )) . Recently, Conditional Value-at-Risk [15] became very popular. CVaR α , α ∈ (0 , 1) being the con-fidence level, is recognized as a model of risk in volatile economic circumstances. For a discreterandom variable q taking on values 0 ≤ v < v < . . . < v ℓ with probabilities q j , 1 ≤ j ≤ ℓ ,Value-at-Risk VaR α and Conditional Value-at-Risk CVaR α are defined by VaR α ( q ) = min v κ : κ X j =1 q j ≥ α and CVaR α ( q ) = 11 − α X κ : v κ ≤ VaR α ( q ) q κ − α · VaR α ( q ) + X κ : v κ > VaR α ( q ) q κ · v κ . .2.1 The Valuation EVar γ EVar γ is a one-parameter valuation represented by the concave function F γ ( x ) := a · x + b · (1 − x ) + γ · ( b − a ) · x · (1 − x ) , with F γ (0) = b and F γ (1) = a . Note that( F γ ( x )) ′ = ( a − b ) + γ · (1 − x ) · ( b − a ) , with ( F γ ( x )) ′ x = x = 0 ⇐⇒ x = 12 (cid:18) − γ · ( b − a ) (cid:19) . Since ( F γ ( x )) ′′ = − γ · ( b − a ) < x ∈ [0 , F γ ( x ) has a local maximumat x = x .To determine the monotonicity properties of F γ ( x ), we distinguish two cases: • For γ · ( b − a ) ≤ F γ ( x ) decreases strictly monotone for x ∈ [0 , • For γ · ( b − a ) > 1, 0 < x < 12 , and F γ ( x ) increases strictly monotone for x ∈ [0 , x ] anddecreases strictly monotone for x ∈ [ x , F γ is a unimodal valuation. Further, note that F γ (cid:16) m (cid:17) = am + m − m · b + γ · ( b − a ) · m − m .Thus, F γ (cid:16) m (cid:17) = b if and only if γ · ( b − a ) = m − m , for m ∈ N . ESD γ ESD γ is a one-parameter valuation represented by the concave function F γ ( x ) := a · x + b · (1 − x ) + γ · ( b − a ) · p x · (1 − x ) , with F γ (0) = b and F γ (1) = a . We have( F γ ( x )) ′ = ( a − b ) + γ · − x p x · (1 − x ) · ( b − a ) . Note that ( F γ ( x )) ′ < x > 12 . Hence, we seek x < 12 with ( F γ ( x )) ′ x = x = 0. So,( F γ ( x )) x = x = 0 ⇐⇒ γ · − x p x · (1 − x ) = 1 ⇐⇒ γ · (1 − x ) = 2 p x · (1 − x ) . It follows that γ · (1 − x + 4 x ) = 4( x − x ), yielding x = 4( γ + 1) − p γ + 1) − γ ( γ + 1)2 · γ + 1) = 12 − p γ + 1 . Since γ > 0, it follows that 0 < x < 12 . Since ( F γ ( x )) ′ x =0 = + ∞ and ( F γ ) ′ x =1 < 0, it followsthat x = 12 − p γ is a local maximum of F γ ( x ), which is unique. Hence, ESD γ is a unimodalvaluation for all values of γ > a, b with a < b . Further, note that F γ (cid:16) m (cid:17) = am + m − m · b + γ · ( b − a ) · √ m − m . Thus, F γ (cid:16) m (cid:17) = b if and only if γ = 1 √ m − m ∈ N .10 .2.3 The Valuation CVaR α CVaR α is a one-parameter valuation represented by the concave function F α ( x ):If x < α , then F α ( x ) = 11 − α · (1 − α ) · b = b , and if x ≥ α , then F α ( x ) = 11 − α · (( x − α ) · a +(1 − x ) · b ).Note that VaR α ( x ) = ( a, if x ≥ αb, if x < α So, F α is a continuous function with F α ( x ) = b for 0 ≤ x ≤ α , F α (1) = a and linear for α ≤ x ≤ As it does not have a unique maximum, it is not a unimodal valuation. Furthermore, since F α ( x ) ismonotone decreasing in x , an E -equilibrium is also an F α -equilibrium, but since F α ( x ) is constantfor 0 ≤ x ≤ α , this does not hold vice versa. So, an F α -equilibrium always exists, but computingan F α -equilibrium is not necessarily PPAD -hard.Observe that the WEEP does not hold for F α . To see this, consider the game (cid:18) ( a, b ) ( b, a )( b, a ) ( a, b ) (cid:19) with α = , p (1) = , p (2) = .Then x ( p , p ) = , x ( p , p ) = , and F α ( p , p ) = F α ( p , p ) = b but E ( p , p ) = · a + · b = E ( p , p ) = · a + · b . So F α does not have the WEEP . As we will show shortly, unimodal valuationshave the WEEP .Finally, note that while for 2-players, 2-values games there always exists a CVaR α -equilibrium, thisis not true for 2-players, 3-values games. It is shown in [13, Theorem 6] that the Crawford game (cid:18) (2 , 2) (1 , , 3) (3 , (cid:19) has no CVaR α -equilibrium. We prove some properties of unimodal valuations. First we provide necessary definitions. Considera 2-players, 2-values game with a unimodal valuation V and a mixed profile h p , p i . Then, we saythat player k ∈ [2] is V -constant on σ ( p k ), if V k ( b p k , p ¯ k ) remains constant over all strategies b p k with σ ( b p k ) ⊆ σ ( p k ). The notion of a player being E -constant is defined similarly. We now show: Lemma 3.2 For a 2-players, 2-values game with a unimodal valuation V , consider a mixed profile h p , p i . For each player k ∈ [2] , if k is E -constant on σ ( p k ) or if k is V -constant on σ ( p k ) , then x k ( b p k , p ¯ k ) is constant for all b p k with σ ( b p k ) ⊆ σ ( p k ) . Proof. We consider first the case that player k ∈ [2] is E -constant on σ ( p k ). Since E k ( b p k , p ¯ k ) = E k ( x k ( b p k , p ¯ k )) and the one-parameter function E ( x ) is strictly monotone decreasing in x , thisimplies that x k ( b p k , p ¯ k ) is constant for all b p k with σ ( b p k ) ⊆ σ ( p k ).Now consider the case that player k ∈ [2] is V -constant on σ ( p k ). Assume on the contrary thatthere exist e p k , ≈ p k with σ ( e p k ) ⊆ σ ( p k ) and σ ( ≈ p k ) ⊆ σ ( p k ) such that y = x k ( e p k , p ¯ k ) < x k ( ≈ p k , p ¯ k ) = z. Since V k ( b p k , p ¯ k ) = F k ( x k ( b p k , p ¯ k )) and since F is a concave function with a unique maximum, thisimplies y < x < z where x is the position in which the unique maximum of F is obtained. Theproperties of F imply additionally that F ( b x ) > F ( y ) = F ( z ) for all b x with y < b x < z . Nowconsider the mixed strategy q k = e p k + ≈ p k . It follows that q k ⊂ σ ( p k ) since σ ( p k ) is a convex set.Furthermore, y < x k ( q k , p ¯ k ) < z and therefore F ( x k ( q k , p ¯ k )) > F ( y ) = F ( z ). This contradicts thefact that V k ( b p k , p ¯ k ) = F k ( x k ( b p k , p ¯ k )) is constant on σ ( p k ).11 emark: Since E and V are one-parameter valuations, if x k ( b p k , p ¯ k ) is constant on σ ( p k ) , then E k ( b p k , p ¯ k ) and V k ( b p k , p ¯ k ) are constant on σ ( p k ) . Lemma 3.3 A unimodal valuation V has the Optimal-Value property and the WEEP . Proof. Consider a V -equilibrium h p , p i . Since V is concave, V has the Optimal-Value property(Proposition 2.1). Hence, for each player k ∈ [2], V k ( b p k , p k ) remains constant over all strategies b p k with σ ( b p k ) ⊆ σ ( p k ). According to Lemma 3.2 and the remark following Lemma 3.2, this impliesthat E k ( b p k , p k ) is constant on σ ( p k ). The WEEP follows.As a special case, Lemma 3.2 immediately implies: Corollary 3.4 For a 2-players, 2-values game G with a unimodal valuation V , consider the mixedprofile h p , p i . For each player k ∈ [2] , if | σ ( p k ) | = n , then k is E -happy with h p , p i if and onlyif k is V -happy with h p , p i . Corollary 3.4 immediately implies: Corollary 3.5 A 2-players, 2-values, 2-strategies game with a unimodal valuation V has a V -equilibrium. We now prove a necessary condition for the existence of an F -equilibrium, which we shall later userepeatedly in the proofs of Theorems 4.1 and 4.3: Lemma 3.6 Consider a 2-players, 2-values game G with a unimodal valuation F , and a mixedprofile h p , p i with the following three properties: (1) There is a strategy b i ∈ σ ( p ) such that µ ( b i, j ) = b for all j ∈ σ ( p ) . (2) For each strategy j ∈ σ ( p ) , there is a strategy i ∈ S with µ ( i, j ) = a . (3) It holds that F ( x ) < b for all x ≥ | σ ( p ) | .Then, p is not an F -best response to p . Note that Property (2) is always fulfilled when G is a normal game (Definition 2.3) due to Condition (2) . Proof. Assume, by way of contradiction, that p is an F -best response to p . By Lemma 3.3, V has the WEEP . Hence, by Property (1) , Lemma 2.2 implies that µ ( i, j ) = µ ( b i, j ) = b for all( i, j ) ∈ σ ( p ) × σ ( p ). It follows that x ( p , p ) = 0. Hence, F ( x ( p , p )) = F (0) = b .By Property (2) , for each strategy j ∈ σ ( p ), there is a strategy i ∈ S with µ ( i, j ) = a ;call it i ( j ). Thus, for each strategy j ∈ σ ( p ), i ( j ) σ ( p ). Set y := max { p ( j ) | j ∈ σ ( p ) } .Then, y ≥ | σ ( p ) | . Choose e j ∈ σ ( p ) with p ( e j ) = y , and set e i := i ( e j ). Then, V (cid:16) p e i , p (cid:17) = F (cid:16) x ( p e i , p ) (cid:17) = F ( x ), for some x with x ≥ y ≥ | σ ( p ) | . It follows, by Property (3) , that F ( x ) < b .So player 1 improves by switching to strategy e i . A contradiction.We shall sometimes use Lemma 3.6 with the roles of the two players interchanged.12 .4 The case x ( F ) = 0 We observe: Lemma 3.7 Assume that the unimodal valuation F is strictly monotone decreasing, and consider amixed profile p = h p , p i . Then, p is an E -best-response to p if and only if it is an F -best-responseto p . Proof. Consider a mixed strategy b p for player 1. Since both E and F are monotone decreasing in x , we have that F ( x ( b p , p )) < F ( x ( p , p )) ⇐⇒ x ( b p , p ) < x ( p , p ) ⇐⇒ E ( x ( b p , p )) < E ( x ( p , p )) . Hence, p is an E -best-response to p if and only if it is an F -best-response to p .Since E -equilibria are invariant to translating and scaling the cost values a and b , Lemma 3.7implies that for a strictly monotone decreasing F , computing an F -equilibrium for a 2-values gamesis as hard as computing an E -equilibrium for a win-lose game, where a = 0 and b = 1. The latteris PPAD -hard [1]. Hence, computing an F -equilibrium for a strictly monotone decreasing F is PPAD -hard. In the case x ( F ) = 0, F is strictly monotone decreasing for x ∈ [0 , Theorem 3.8 For a unimodal valuation F , if x ( F ) = 0 , then an F -equilibrium exists for all 2-players, 2-values games and its computation is PPAD -hard. Recall that for γ · ( b − a ) ≤ EVar γ is strictly monotone decreasing for x ∈ [0 , 1] and EVar γ (0) = b .Hence, x ( EVar γ ) = 0. Therefore: Corollary 3.9 Computing an EVar γ -equilibrium, with γ · ( b − a ) ≤ , is PPAD -hard for 2-players,2-values games. We begin with a property about uniqueness of equilibria which will have the essential role in theproof of the non-existence result. For this purpose, we introduce the 2-values, m -strategies bimatrixgame D m , with m ≥ D m = ( a, b ) ( b, a ) ( b, b ) . . . ( b, b ) ( b, b )( b, b ) ( a, b ) ( b, a ) . . . ( b, b ) ( b, b )... ... ... . . . ... ...( b, b ) ( b, b ) ( b, b ) . . . ( a, b ) ( b, a )( b, a ) ( b, b ) ( b, b ) . . . ( b, b ) ( a, b ) . Thus, D m = ( α ij , β ij ) ≤ i,j ≤ m − is a Toeplitz bimatrix, with: α ij = (cid:26) a , if i = jb , otherwise and β ij = (cid:26) a , if j = ( i + 1) mod mb , otherwise . Clearly, D m is a normal game. Thus: 13 For each strategy j ∈ { , , . . . , m − } , there is a strategy i ∈ { 0, 1, . . . , m-1 } with α ij = a .This implies that any mixed profile h p , p i fulfills Property (2) from Lemma 3.6.Note that for each player, there is exactly one a in every row and every column. This property isstronger than Conditions (2) and (3) together in the definition of a normal game. We show: Theorem 4.1 Consider a unimodal valuation F . D m has the F -equilibrium h p , p i , given by p ( j ) = p ( j ) = 1 m , for ≤ j ≤ m − . D m has no other F -equilibrium if and only if one ofthe following three conditions holds: ( i ) m ≤ , ( ii ) m is even, m ≥ , F (cid:0) m (cid:1) = b and F ( x ) < b for x ≥ m − , ( iii ) m is odd, m ≥ and F ( x ) < b for x ≥ m − . Before proceeding to the proof, we present a property concerning the distribution of the b -values. Definition 4.1 ( b -blocks and b -double-blocks) Let A, B ⊆ { , . . . , m − } with | A | ≥ and | B | ≥ . Then, ( i ) ( A, B ) is a b -block for player 1 if α ij = b for all i ∈ A, j ∈ B . ( ii ) ( A, B ) is a b –double-block if α ij = β ij = b for all i ∈ A, j ∈ B .In a corresponding way, a b -block for player 2 is defined. We show: Lemma 4.2 Let A, B ⊆ { , . . . , m − } . ( i ) If ( A, B ) is a b -block for some player k ∈ [2] , then | A | + | B | ≤ m . ( ii ) If ( A, B ) is a b -double-block, then | A | + | B | ≤ m − . Proof. Note that | B | > | A | < m and | A | > | B | < m .( i ) We do the proof only for player 1. Note that α ii = a for all i ∈ { } . So, if i ∈ A then i / ∈ B . This implies | B | ≤ m − | A | , and therefore | A | + | B | ≤ m .( ii ) Note that for all i ∈ { , . . . , m − } , α ii = β i, ( i +1)mod m = a . So, if i ∈ A , then i, ( i +1) mod m / ∈ B . Since | A | < m , |{ i, ( i + 1) mod m : i ∈ A }| ≥ | A | + 1. This implies | B | ≤ m − ( | A | + 1),and therefore | A | + | B | ≤ m − Proof. By the construction of D m , for each player k ∈ [2] and for each strategy i ∈ { , . . . , m − } , E k ( p i , p ) = a · m + b · m − m , which is independent of i . Thus, h p , p i has the WEEP . Since h p , p i is fully mixed, it follows that h p , p i is an E -equilibrium. Hence, by Corollary 3.4, h p , p i is an F -equilibrium.We now prove that the stated conditions are necessary and sufficient for uniqueness. First we showthat they are sufficient. “ ⇐ ”: Consider an arbitrary F -equilibrium h p , p i . We distinguish three cases.14 ( σ ( p ) , σ ( p )) is a b -double-block: Due to the structure of D m , such a double-block of b ’s doesnot exist if m ≤ 4. Now let m ≥ 5. Due to Lemma 4.2, | σ ( p ) | + | σ ( p ) | ≤ m − 1. Due to thesymmetry of D m , we can assume that | σ ( p ) | ≤ | σ ( p ) | . Then, | σ ( p ) | ≤ ⌊ m − ⌋ . We have todistinguish even and odd m .– If m is even, then | σ ( p ) | ≤ m − , and by assumption F ( x ) < b for x ≥ m − . Therefore, F ( x ) < b for x ≥ | σ ( p ) | . Lemma 3.6 implies that p is not an F -best response to p . Acontradiction.– If m is odd, then | σ ( p ) | ≤ m − , and by assumption F ( x ) < b for x ≥ m − . Therefore, F ( x ) < b for x ≥ | σ ( p ) | . Lemma 3.6 implies that p is not an F -best response to p . Acontradiction. (2) For some k ∈ [2], ( σ ( p ) , σ ( p )) is a b -block for player k but not a b -block: Due to the sym-metry of D m , we can assume that k = 2. Then there is a pair (ˆ i, ˆ j ) ∈ σ ( p ) × σ ( p ) with α ˆ i ˆ j = a. We first prove that for every i ∈ σ ( p ) , there is some j ∈ σ ( p ) with α ij = a .Assume, by way of contradiction, that α e i,j = b for some e i ∈ σ ( p ) and all j ∈ σ ( p ) . Then,for player 1, strategy b i ∈ σ ( p ) dominates strategy e i ∈ σ ( p ) with respect to σ ( p ) . By Lem-mas 2.2 and 3.3, it follows that no strategy in σ ( p ) dominates some other strategy in σ ( p )with respect to σ ( p ). A contradiction.Note that for all i, j ∈ { , . . . , m − } , α ij = a if and only if j = i . So, i ∈ σ ( p ) implies i ∈ σ ( p ). Therefore, | σ ( p ) | ≤ | σ ( p ) | . Lemma 4.2 implies | σ ( p ) | + | σ ( p ) | ≤ m , andtherefore | σ ( p ) | ≤ ⌊ m ⌋ . We distinguish even and odd m .– If m is odd, then | σ ( p ) | ≤ m − , and by assumption F ( x ) < b for x ≥ m − . Therefore, F ( x ) < b for x ≥ | σ ( p ) | . Lemma 3.6 implies that p is not an F -best response to p . Acontradiction.– If m is even, then | σ ( p ) | ≤ m . We have two subcases, | σ ( p ) | < m and | σ ( p ) | = m .+ | σ ( p ) | ≤ m − 1: By assumption, F ( x ) < b for x ≥ m − . Therefore, F ( x ) < b for x ≥ | σ ( p ) | . Lemma 3.6 implies that p is not an F -best response to p . Acontradiction.+ | σ ( p ) | = m : By assumption, F (cid:0) m (cid:1) = b . If F (cid:0) m (cid:1) < b , then Lemma 3.6 implies that p is not an F -best response to p . A contradiction. If F (cid:0) m (cid:1) > b , then the WEEP implies that p ( j ) = m for all j ∈ σ ( p ). Thus V ( p , p ) = F (cid:0) m (cid:1) > b and henceplayer 1 can improve by using some strategy i / ∈ σ ( p ). A contradiction. (3) ( σ ( p ) , σ ( p )) is neither a b -block for player 1 nor a b -block for player 2: Then, there is a pair(ˆ i, ˆ j ) ∈ σ ( p ) × σ ( p ) with α ˆ i ˆ j = a and there is a pair ( e i, e j ) ∈ σ ( p ) × σ ( p ) with β e i e j = a .Since in an equilibrium no strategy can dominate some other (Lemmas 2.2 and 3.3), for every i ∈ σ ( p ) there is some j ∈ σ ( p ) with α ij = a and for every j ∈ σ ( p ) there is some i ∈ σ ( p )with β ij = a . A more elaborate proof was given in case (2) .Due to the structure of ( α ij ) ≤ i,j ≤ m − , it follows that for every i ∈ σ ( p ), we also have i ∈ σ ( p ). Due to the structure of ( β ij ) ≤ i,j ≤ m − , it follows that for every j ∈ σ ( p ), we also have( j − m ) ∈ σ ( p ). These two taken together yield that σ ( p ) = σ ( p ) = { , . . . , m − } .15y the WEEP for player 1, the expression a · p ( i ) + b · X i ′ ∈{ ,...,m − }\{ i } p ( i ′ )is constant for i ∈ { , . . . , m − } , yielding the unique solution p ( i ) = 1 m for each strategy i ∈ { 0, . . . , m-1 } . By the WEEP for player 2, we get identically the unique solution p ( j ) = 1 m for each strategy j ∈ { 0, . . . , m-1 } . “ ⇒ ”: We now show that the conditions are necessary for uniqueness. The conditions do not holdif and only if (there exists even m ≥ F (cid:0) m (cid:1) = b or F (cid:16) m − (cid:17) ≥ b ) or (there exists odd m ≥ F (cid:16) m − (cid:17) ≥ b ). (1) If m is even, m ≥ F (cid:0) m (cid:1) = b , then σ ( p ) = { i ∈ { , . . . , m − } : i even } , p ( i ) = m for all i ∈ σ ( p ) σ ( p ) = { j ∈ { , . . . , m − } : j odd } , p ( j ) = m for all j ∈ σ ( p )is an F -equilibrium. | σ ( p ) | = | σ ( p ) | = m , and p is uniform on σ ( p k ) for k ∈ [2].For example, for m = 4 and F (cid:0) (cid:1) = b we get: ( a, b ) (b,a) ( b, b ) (b,b)( b, b ) ( a, b ) ( b, a ) ( b, b )( b, b ) (b,b) ( a, b ) (b,a)( b, a ) ( b, b ) ( b, b ) ( a, b ) . (2) If m is even, m ≥ F (cid:16) m − (cid:17) ≥ b , then σ ( p ) = { i : m ≤ i ≤ m − } , p ( i ) = m − for all i ∈ σ ( p ) σ ( p ) = { j : 0 ≤ j ≤ m − } , p ( j ) = m − for all j ∈ σ ( p )is an F -equilibrium. | σ ( p ) | = | σ ( p ) | = m − , and p is uniform on σ ( p k ) for k ∈ [2].For example, for m = 6 and F (cid:0) (cid:1) ≥ b we get: ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( b, b ) ( b, b )( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( b, b )( b, b ) ( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b )(b,b) (b,b) ( b, b ) ( a, b ) ( b, a ) ( b, b )(b,b) (b,b) ( b, b ) ( b, b ) ( a, b ) ( b, a )( b, a ) ( b, b ) ( b, b ) ( b, b ) ( b, b ) ( a, b ) . (3) If m is odd, m ≥ F (cid:16) m − (cid:17) ≥ b , then σ ( p ) = (cid:8) i : m − ≤ i ≤ m − (cid:9) , p ( i ) = m − for all i ∈ σ ( p ) σ ( p ) = (cid:8) j : 0 ≤ j ≤ m − (cid:9) , p ( i ) = m − for all j ∈ σ ( p )16s an F -equilibrium. | σ ( p ) | = | σ ( p ) | = m − , and p is uniform on σ ( p k ) for k ∈ [2].For example, for m = 5 and F (cid:0) (cid:1) ≥ b we get: ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( b, b )( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b )(b,b) (b,b) ( a, b ) ( b, a ) ( b, b )(b,b) (b,b) ( b, b ) ( a, b ) ( b, a )( b, a ) ( b, b ) ( b, b ) ( b, b ) ( a, b ) . This completes the proof.We now proceed to prove the inexistence result. Game C m is a 2-values, ( m + 1)-strategies bimatrixgame, with m ≥ 2, which is derived by adding one row and one column to game D m , as follows: C m = ( a, b )( b, b ) D m ...( b, b )( b, b ) ( b, b ) . . . ( b, b ) ( b, b ) ( b, a ) . Thus, C m = ( α ij , β ij ) ≤ i,j ≤ m , where: α ij , β ij for 0 ≤ i, j ≤ m − D m , α mj = α im = b for 1 ≤ i ≤ m, ≤ j ≤ m , α m = a , β mj = β im = b for 0 ≤ i ≤ m − , ≤ j ≤ m − 1, and β mm = a .Clearly, C m is a normal game. Due to the structure of C m , there hold two properties with corre-sponding consequences to Properties (1) and (2) from Lemma 3.6: • α mj = b for all j ∈ { , , . . . , m } . Thus, µ ( m, j ) = b for all j ∈ { , , . . . , m } . Taking b i := m ,this implies Property (1) from Lemma 3.6 for any mixed profile ( p , p ) with m ∈ σ ( p ). β im = b for all i ∈ { , . . . , m − } . Thus, µ ( i, m ) = b for all i ∈ { , . . . , m − } . Taking b j := m ,this implies Property (1) from Lemma 3.6 for any mixed profile ( p , p ) with m ∈ σ ( p ) and m σ ( p ). • Since C m is a normal game, any mixed profile ( p , p ) fulfills Property (2) from Lemma 3.6.We show: Theorem 4.3 Let m ≥ . Consider a unimodal valuation F . C m has no F -equilibrium if and onlyif F (cid:16) m (cid:17) > b and one of the following conditions holds: i ) m is even, F ( x ) < b for x ≥ m , ( ii ) m is odd, F (cid:16) m +1 (cid:17) = b and F ( x ) < b for x ≥ m − . Proof. We first prove that the conditions are sufficient for non-existence. In the proof we will makeuse of the following observation: if F ( z ) < b then F ( x ) decreases strictly monotone for x ∈ ( z, F ( z ) < b and z < x , then F ( x ) < b . “ ⇐ ”: Assume, by way of contradiction, that C m has an F -equilibrium h p , p i with m σ ( p )and m σ ( p ). Due to the structure of C m , it follows that h p , p i is an F -equilibrium for D m .Since F ( x ) < b for x ≥ m , with m even, or x ≥ m − , with m odd, by the above observationand Theorem 4.1, it follows that p ( j ) = p ( j ) = m with 0 ≤ j ≤ m − 1. This implies that x ( p , p ) = m . Thus, V ( p , p ) = F (cid:16) m (cid:17) > b . Then, player 1 could improve by switching tostrategy m with V ( p m , p ) = b . A contradiction.Assume now that there is an F -equilibrium h p , p i with m ∈ σ ( p ) or m ∈ σ ( p ). Since game C m is normal, as mentioned above, Property (2) from Lemma 3.6 is fulfilled. There remain two casesto consider. (1) m ∈ σ ( p ): Then ( σ ( p ) \{ m } , σ ( p ) \{ m } ) is a b -block for player 1. This leads to two subcases: (1.1) ( σ ( p ) \ { m } , σ ( p ) \ { m } ) is a b -double-block: This implies that m / ∈ σ ( p ), hence | σ ( p ) \ { m }| = | σ ( p ) | . From Lemma 4.2 we have that | σ ( p ) \ { m }| + | σ ( p ) \ { m }| ≤ m − 1. Hence, | σ ( p ) | + | σ ( p ) | ≤ m . Let k ∈ [2] with | σ ( p k ) | ≤ | σ ( p ¯ k ) | . Then | σ ( p k ) | ≤ ⌊ m ⌋ . If m is even (resp., odd), then | σ ( p k ) | ≤ m and by assumption F ( x ) < b for x ≥ m (resp., | σ ( p k ) | ≤ m − and by assumption F ( x ) < b for x ≥ m − ). In bothcases, F ( x ) < b for x ≥ | σ ( p k ) | . Lemma 3.6 implies that p ¯ k is not an F -best response to p k . A contradiction. (1.2) ( σ ( p ) \ { m } , σ ( p ) \ { m } ) is not a b -block for player 2: Then, for every j ∈ σ ( p ) thereexists i ∈ σ ( p ) with β ij = a . For j ∈ σ ( p ) \ { m } we get that ( j − 1) mod m ∈ σ ( p ) \ { m } and hence | σ ( p ) \ { m }| ≤ | σ ( p ¯1 ) \ { m }| . Lemma 4.2 implies that | σ ( p ) \{ m }| + | σ ( p ) \ { m }| ≤ m . We consider two subcases: (1.2.1) m / ∈ σ ( p ): Then | σ ( p ) | + | σ ( p ) | ≤ m +1 and | σ ( p ) | ≤ | σ ( p ) |− . Thus, 2 ·| σ ( p ) | ≤| σ ( p ) | + | σ ( p ) | − ≤ m , which yields | σ ( p ) | ≤ ⌊ m ⌋ . Following the exact samereasoning as in case (1.1) and using Lemma 3.6, we conclude that p is not an F -bestresponse to p . A contradiction. (1.2.2) m ∈ σ ( p ): Because of the non-symmetry of C m we now need to employ new ideas.Observe that m ∈ σ ( p ) implies 0 / ∈ σ ( p ) (since α m = a and ( σ ( p ) \ { m } , σ ( p ) \{ m } ) is a b -block for player 1). We distinguish two subcases: (1.2.2.1) / ∈ σ ( p ): For i ∈ { , . . . , m − } , i ∈ σ ( p ) implies i / ∈ σ ( p ). Hence, ( σ ( p ) \{ m } ) ∩ ( σ ( p ) \ { m } ) = ∅ . Furthermore, σ ( p ) \ { m } ⊆ { , . . . , m − } , σ ( p \{ m } ⊆ { , . . . , m − } implies | σ ( p ) \ { m }| + | σ ( p ) \ { m }| ≤ m − 1, whichimplies | σ ( p ) | + | σ ( p ) | ≤ m + 1, and hence | σ ( p ) | ≤ ⌊ m +12 ⌋ . We distinguishbetween m even and m odd.– If m is even, then | σ ( p ) | ≤ m and by assumption F ( x ) < b for x ≥ m . Hence, F ( x ) < b for x ≥ | σ ( p ) | . Lemma 3.6 implies that p is not an F -best responseto p . A contradiction. 18 If m is odd, then | σ ( p ) | ≤ m +12 . We have two subcases, | σ ( p ) | < m +12 and | σ ( p ) | = m +12 .+ | σ ( p ) | ≤ m +12 − 1: By assumption, F ( x ) < b for x ≥ m − . Therefore, F ( x ) < b for x ≥ | σ ( p ) | . Lemma 3.6 implies that p is not an F -bestresponse to p . A contradiction.+ | σ ( p ) | = m +12 : By assumption, F (cid:16) m +1 (cid:17) = b . If F (cid:16) m +1 (cid:17) < b , thenLemma 3.6 implies that p is not an F -best response to p . A contradic-tion. For F (cid:16) m +1 (cid:17) > b , the WEEP implies that p ( i ) = m for all i ∈ σ ( p ).Thus V ( p , p ) = F (cid:0) m (cid:1) > b and hence player 2 can improve by using somestrategy j / ∈ σ ( p ). A contradiction. (1.2.2.2) ∈ σ ( p ): Then, | σ ( p ) \ { m }| + | σ ( p ) \ { m }| ≤ m , which implies, | σ ( p ) | + | σ ( p ) | ≤ m + 2, and hence | σ ( p ) | ≤ ⌊ m +22 ⌋ . Unlike the previous cases, we donot get the optimal result by just applying Lemma 3.6. Instead, we will show,by using the method in the proof of Lemma 3.6, that p is not an F -best responseto p if F ( x ) < b for x ≥ | σ ( p ) |− , deriving a contradiction.For every j ∈ { , . . . , m } there is exactly one strategy i ∈ { , . . . , m − } with α ij = a , call it i ( j ). Then, for every strategy j ∈ σ ( p ) , i ( j ) ∈ σ ( p ). It is i (0) = i ( m ) = 0 and i ( j ) = i ( j ) for j = j = 0. Then, V ( p , p ) = F ( p (0) + p ( m ))and for j ∈ σ ( p ) \ { , m } , V ( p i ( j )1 , p ) = F ( p ( j )). Then, p (0) + p ( m ) ≥ | σ ( p ) |− or there exists some j ∈ σ ( p ) \ { , m } with p ( j ) ≥ | σ ( p ) |− . So, when F ( x ) < b for x ≥ | σ ( p ) |− , player 1 can improve by switching either to strategy0 or to a strategy i ( j ) with p ( j ) ≥ | σ ( p ) |− . Recall that | σ ( p ) | ≤ ⌊ m +22 ⌋ . So,player 1 can improve for F (cid:16) ⌊ m ⌋ (cid:17) < b . A contradiction.To better understand this case, we provide an example with m = 4. ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( a, b )(b,b) ( a, b ) (b,a) ( b, b ) (b,b)( b, b ) ( b, b ) ( a, b ) ( b, a ) ( b, b )(b,a) ( b, b ) (b,b) ( a, b ) (b,b)(b,b) ( b, b ) (b,b) ( b, b ) (b,a) Then, σ ( p ) = { , , } , σ ( p ) = { , , } , p (0) + p (4) ≥ or p (2) ≥ . For F (cid:0) (cid:1) < b , player 1 can improve by switching to strategy 0 or to strategy 2. (2) m / ∈ σ ( p ), m ∈ σ ( p ): Then ( σ ( p ) , σ ( p ) \ { m } ) is a b -block for player 2. We distinguish twocases: (2.1) ( σ ( p ) , σ ( p ) \ { m } ) is a b -double-block: From Lemma 4.2 we have that | σ ( p ) | + | σ ( p ) \{ m }| ≤ m − 1. Hence, | σ ( p ) | + | σ ( p ) | ≤ m . Let k ∈ [2] with | σ ( p k ) | ≤ | σ ( p ¯ k ) | .Then | σ ( p k ) | ≤ ⌊ m ⌋ . Following exactly the same reasoning as in case (1.1) and usingLemma 3.6, we conclude that p ¯ k is not an F -best response to p k . A contradiction. (2.2) ( σ ( p ) , σ ( p ) \ { m } ) is not a b -block for player 1: Then, for every i ∈ σ ( p ) there exists j ∈ σ ( p ) with α ij = a , i.e., for i ∈ σ ( p ) \ { } , we get that i ∈ σ ( p ). This implies that σ ( p ) \ { } ⊆ σ ( p ) \ { m } , and hence | σ ( p ) | ≤ | σ ( p ) | . Lemma 4.2 implies | σ ( p ) | +19 σ ( p ) \ { m }| ≤ m , which implies | σ ( p ) | + | σ ( p ) | ≤ m + 1, and hence | σ ( p ) | ≤ ⌊ m +12 ⌋ .Following exactly the same reasoning as in case (1.2.2.1) , we reach a contradiction. “ ⇒ ”: We now show that the conditions are necessary, that is, if they do not hold, then there is an F -equilibrium. We first show the necessity of F (cid:0) m (cid:1) > b . If F (cid:0) m (cid:1) ≤ b then C m has an F -equilibrium h p , p i with σ ( p ) = { i ∈ { , . . . , m − }} , p ( i ) = m for all i ∈ σ ( p ) σ ( p ) = { j ∈ { , . . . , m − }} , p ( j ) = m for all j ∈ σ ( p ) | σ ( p ) | = | σ ( p ) | = m , and p is uniform on σ ( p k ) for k ∈ [2] (this is the case m / ∈ σ ( p ) and m / ∈ σ ( p )).Then, given that F (cid:0) m (cid:1) > b , the remaining conditions do not hold if and only if (there exists m even with F (cid:0) m (cid:1) ≥ b ) or (there exists m odd with F (cid:16) m +1 (cid:17) = b or F (cid:16) m − (cid:17) ≥ b ). (1) For m even and F (cid:0) m (cid:1) ≥ b , C m has an F -equilibrium h p , p i with σ ( p ) = { m } ∪ { i : m ≤ i ≤ m − } ,σ ( p ) = { j : 0 ≤ j ≤ m − } , | σ ( p ) | = | σ ( p ) | = m , and p is uniform on σ ( p k ) for k ∈ [2].Observe that we are in case (1.1) : m ∈ σ ( p ) , m / ∈ σ ( p ), and ( σ ( p ) \ { m } , σ ( p ) \ { m } ) is a b -double-block.For example, for m = 4 and the above case, C = ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( a, b )( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b )(b,b) (b,b) ( a, b ) ( b, a ) ( b, b )( b, a ) ( b, b ) ( b, b ) ( a, b ) ( b, b )(b,b) (b,b) ( b, b ) ( b, b ) ( b, a ) . (2) For m odd and F (cid:16) m +1 (cid:17) = b , C m has an F -equilibrium h p , p i with σ ( p ) = { i ∈ { , . . . , m } : i odd } ,σ ( p ) = { m } ∪ { j ∈ , . . . , m − } : j even } , | σ ( p ) | = | σ ( p ) | = m +12 , and p is uniform on σ ( p k ) for k ∈ [2].Observe that we are in case (1.2.2.1) : m ∈ σ ( p ) , m ∈ σ ( p ) , and 0 / ∈ σ ( p ).For example, for m = 5 and the above case, C = ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( b, b ) ( a, b )( b, b ) ( a, b ) (b,a) ( b, b ) (b,b) (b,b)( b, b ) ( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b )( b, b ) ( b, b ) (b,b) ( a, b ) (b,a) (b,b)( b, a ) ( b, b ) ( b, b ) ( b, b ) ( a, b ) ( b, b )( b, b ) ( b, b ) (b,b) ( b, b ) (b,b) (b,a) . (3) For m odd and F (cid:16) m − (cid:17) ≥ b , C m has an F -equilibrium h p , p i with20 ( p ) = { m } ∪ { i : m +12 ≤ i ≤ m − } ,σ ( p ) = { j : 0 ≤ j ≤ m − } , | σ ( p ) | = | σ ( p ) | = m − , and p is uniform on σ ( p k ) for k ∈ [2].Observe that we are again in case (1.1) : m ∈ σ ( p ) , m / ∈ σ ( p ), and ( σ ( p ) \ { m } , σ ( p ) \ { m } )is a b -double-block.For example, for m = 5 and the above case, C = ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( b, b ) ( a, b )( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b ) ( b, b )( b, b ) ( b, b ) ( a, b ) ( b, a ) ( b, b ) ( b, b )(b,b) (b,b) ( b, b ) ( a, b ) ( b, a ) ( b, b )( b, a ) ( b, b ) ( b, b ) ( b, b ) ( a, b ) ( b, b )(b,b) (b,b) ( b, b ) ( b, b ) ( b, b ) ( b, a ) . This completes the proof.Now, putting together the results of Theorems 3.8 and 4.3, we obtain a panorama on the(in)existence of F -equilibria, summarized in the following theorem: Theorem 4.4 For a unimodal valuation F the following properties hold: ( i ) If x ( F ) = 0 then an F -equilibrium exists for all 2-players, 2-values games and its computationis PPAD -hard. ( ii ) If x ( F ) > and F (cid:16) (cid:17) = b , then there exists a normal 2-players, 2-values game without F -equilibrium. Proof. Property ( i ) follows directly from Theorem 3.8. Now consider the case x ( F ) > F (cid:0) (cid:1) = b . Then, x = x ( F ) exists with F ( x ) = b . Choose m ∈ N with m < x ( F ) ≤ m − . Then, F (cid:0) m (cid:1) > b , F (cid:16) m − (cid:17) ≤ b and F ( x ) decreases strictly monotone for x ∈ h m − , i . We distinguishbetween m even and m odd. • If m ≥ m > m − and hence F (cid:0) m (cid:1) < F (cid:16) m − (cid:17) ≤ b . For m = 2, we have F (cid:0) m (cid:1) = F (1) = a < b . • If m ≥ F (cid:16) m − (cid:17) < F (cid:16) m − (cid:17) ≤ b. Furthermore, F (cid:16) m +1 (cid:17) = b (otherwise, x ( F ) = m +1 > m − , contradicting the choice of m s.t. x ( F ) ≤ m − ). For m = 3 we have F (cid:16) m − (cid:17) = F (1) = a < b ; furthermore, m +1 = , and by assumption, F (cid:0) (cid:1) = b .So, for all m ≥ 2, the conditions of Theorem 4.3 hold, and hence the game C m has no F -equilibrium.Theorem 4.4 leaves open the case on whether there exists, or not, an F -equilibrium when F (cid:0) (cid:1) = b .This is the subject of Section 5. 21 Existence of Equilibria In this section we show the existence of equilibria for a 2-players, 2-values game G when G is anormal game with F (cid:0) (cid:1) = b (Section 5.1) and when players have three strategies (Section 5.2),complementing the investigation of the previous section. F (cid:0) (cid:1) = b We now consider normal games. Specifically, in this section G is a normal 2-players game with twovalues a, b, a < b . Furthermore, we assume that F (cid:0) (cid:1) = b .Recall that in a normal game, for each row i there is exactly one column j with β ij = a , andfor each column j there is exactly one row i with α ij = a . For convenience of presentation, in thissection we define the set of strategies over the set [ n ] = { , . . . , n } . Then, for i ∈ [ n ] let col ( i ) be theuniquely determined column with β icol ( i ) = a . For j ∈ [ n ] let row ( j ) be the uniquely determinedrow with α row ( j ) j = a . Set: C = { col ( i ) : i ∈ [ n ] } and R = { row ( j ) : j ∈ [ n ] } . Note that in a normal game, always | C | ≥ , | R | ≥ | C | = 1;then ∃ j ∀ i : γ ( i, j ) = ( b, a ). But in a normal game γ ( row ( j ) , j ) = ( a, b ), a contradiction. In thesame way | R | ≥ G always has an F -equilibrium whose supportsets both have size 2, and that this equilibrium can be computed in linear time, with respect tothe input of the problem (Theorem 5.7 and Corollary 5.8).For the complexity consideration we use the condensed input form where the game is given bythe the two arrays Col and Row with Col ( i ) = col ( i ) and Row ( i ) = row ( i ) for all i ∈ [ n ]. Usingthis input convention, the sets C and R , represented as Boolean arrays, can be computed in time O ( n ). We will show that a winning pair (c.f. Definition 5.1) can be found in time O ( n ). Notethat if the input is given by the two matrices ( α ij ) ≤ i,j ≤ n , ( β ij ) ≤ i,j ≤ n , then the input needs O ( n )space. In this case, the arrays Col and Row can be computed in time O ( n ) and using afterwardsour algorithm leads to an algorithm for computing a winning pair in time O ( n ). Since the inputis of size O ( n ), this is also a linear time algorithm. We now define the notion of a winning pair. Definition 5.1 (Winning Pair) A pair ( σ , σ ) with σ , σ ⊆ [ n ] , | σ | = | σ | = 2 , σ = { i , i } , σ = { j , j } is called a winning pair if the three following conditions arefulfilled:(i) (cid:18) α i j α i j α i j α i j (cid:19) and (cid:18) β i j β i j β i j β i j (cid:19) have the form (cid:18) a bb a (cid:19) or (cid:18) b aa b (cid:19) or (cid:18) b bb b (cid:19) (ii) row ( j ) = row ( j ) (iii) col ( i ) = col ( i )We first show: Lemma 5.1 Let ( σ , σ ) , σ = { i , i } , σ = { j , j } be a winning pair and define a strategy vector h p , p i by p ,i = p ,i = 1 / , p ,j = p ,j = 1 / . Then, h p , p i is an F -equilibrium. roof. By assumption, F (cid:0) (cid:1) = b and therefore V ( p , p ) = V ( p , p ) = b .If player 1 chooses an arbitrary pure strategy i , then, due to condition ( ii ), player 1 has two b ’sor one a and one b . So V ( p i , p ) = b for all i ∈ [ n ]. Hence, player 1 cannot improve by choosingstrategy i . Likewise, due to condition ( iii ), player 2 cannot improve by choosing an arbitrary purestrategy j .In the following lemmas we will show results about the distribution of ( a, b ) entries and ( b, a ) entriesin the case where there exists no winning pair. Lemma 5.2 Let the two pairs σ = { i , i } , σ = { j , j } fulfill the following properties:(i) { i , i } ∩ { row ( j ) , row ( j ) } = ∅{ j , j } ∩ { col ( i ) , col ( i ) } = ∅ (ii) row ( j ) = row ( j ) (iii) col ( i ) = col ( i ) Then, σ × σ is a winning pair. Proof. Observe that properties ( ii ) and ( iii ) are equal to the conditions ( ii ) and ( iii ) of Defini-tion 5.1. Therefore, only condition ( i ) has to be verified. • α row ( j v ) j v = a for v ∈ { , } and normality implies that α ij v = b for all i = row ( j v ). So thefirst line of property ( i ) implies that α i µ j v = b for µ, v ∈ { , } . • β i v col ( i v ) = a for v ∈ { , } and normality implies that β i v j = b for all j = col ( i v ). So thesecond line of property ( i ) implies that β i µ j v = b for µ, v ∈ { , } .The proof is demonstrated in Figure 1. row ( j ) row ( j ) i i j j col ( i ) col ( i )( a, b ) ( a, b ) ( b, a ) ( b, a )( b, b ) ( b, b )( b, b ) ( b, b ) Figure 1: Demonstrating the proof of Lemma 5.2.23 emma 5.3 There exists an algorithm running in time O ( n ) with the following input-output rela-tion.(i) input: j , j ∈ [ n ] with row ( j ) = row ( j ) output:(i/1) column j ∈ [ n ] such that { col ( i ) : i = row ( j ) , i = row ( j ) } ⊂ { j , j , j } , or(i/2) rows i , i ∈ [ n ] , i = i , such that { i , i } × { j , j } is a winning pair.(ii) input: i , i ∈ [ n ] with col ( i ) = col ( i ) output:(ii/1) row i ∈ [ n ] such that { row ( j ) : j = col ( i ) , j = col ( i ) } ⊂ { i , i , i } , or(ii/2) columns j , j ∈ [ n ] , j = j , such that { i , i } × { j , j } is a winning pair. Proof. Due to the symmetry of rows and columns, it is sufficient to prove part ( i ).We describe the algorithm, that starts with the input j , j with row ( j ) = row ( j ): • It checks col ( i ), i ∈ [ n ]. • If ∃ j ∈ [ n ] with { col ( i ) : i = row ( j ) , i = row ( j ) } ⊂ { j , j , j } then it exists ( i/ 1) andoutputs j . • Otherwise the algorithm has found some A ⊆ [ n ] with | A | = 2, A ∩ { j , j } = ∅ and A ⊆{ i : i = row ( j ) , i = row ( j ) } . Take some i , i with A = { col ( i ) , col ( i ) } . Then col ( i ) = col ( i ). Due to Lemma 5.2, { i , i } × { j , j } is a winning pair. Therefore, thealgorithm exits ( i/ 2) and outputs i , i .The algorithm makes its decision following just one scan of col ( i ), i ∈ [ n ]. So, the time bound is O ( n ). Lemma 5.4 There exists an algorithm that computes a winning pair in time O ( n ) provided that n ≥ and ( | R | ≥ or | C | ≥ . Proof. Due to the symmetry of rows and columns, it is sufficient to describe the algorithm in thecase that | R | ≥ • The algorithm scans row ( j ) , j ∈ [ n ] and determines j v , v ∈ [4], with |{ row ( j v ) : v ∈ [ n ] }| = 4. • Then it simulates the algorithm described in the proof of Lemma 5.3 for the six 2-sets { j v , j µ } , v = µ, v, µ ∈ [4].Overall, the algorithm runs in time O ( n ). In the remainder of the proof we will show that in thecase n ≥ 5, for at least one of the pairs { j v , j µ } exit ( i/ 2) is reached, and thus a winning pair isfound. 24e will do so, by proof by contradiction: assume that n ≥ { j v , j µ } exit( i/ 1) is reached.Let j v , v ∈ [4], with |{ row ( j v ) : v ∈ [4] }| = 4. We apply Lemma 5.3 six times, i.e., foreach pair { j v , j µ } , v = µ, v, µ ∈ [4]. Due to Lemma 5.3, there exist constructs j { v,µ } such that col ( i ) ∈ { j v , j µ , j { v,µ } } for all i 6∈ { row ( j µ ) , row ( j v ) } .First choose i 6∈ { row ( j µ ) : µ ∈ [4] } . To be able to do this, we need the assumption that n ≥ 5. For each 2-set { µ, v } ⊂ [4] we denote with { µ, v } the complement of { µ, v } in [4]. Then,due to Lemma 5.3, col ( i ) ∈ ( { j k : k ∈ { µ, v }} ∪ { j { v,µ } } ) ∩ ( { j k : k ∈ { µ, v }} ∪ { j { v,µ } } ) . Since { µ, v } ∩ { µ, v } = ∅ , this implies that col ( i ) = j { v,µ } = j { v,µ } . Now, since the 2-set { µ, v } ⊂ [4]were chosen arbitrarily, this shows that all the j { v,µ } , { v, µ } ⊂ [4] have the same value. Let thisvalue be ˆ j , i.e., col ( i ) = ˆ j for i 6∈ { row ( j µ ) : µ ∈ [4] } .Now consider i = row ( j ρ ) for some ρ ∈ [4]. Then information about col ( i ) is given by Lemma 5.3using the 3 pairs { µ, v } ⊂ [4], ρ 6∈ { µ, v } , i.e., for all i ∈ [ n ], col ( i ) ∈ \ { µ,v }⊂ [4] \{ ρ } { j µ , j v , ˆ j } = ˆ j, contradicting the fact that | C | ≥ Lemma 5.5 There exists an algorithm that computes a winning pair in time O ( n ) , provided that |{ col ( i ) : i R }| ≥ , and |{ row ( j ) : j C }| ≥ . Proof. We describe the algorithm: it scans col ( i ) , row ( i ) , ≤ i ≤ n , and determines i , i R with col ( i ) = col ( i ) and j , j C with row ( j ) = row ( j ). This takes O ( n ) time. Since i , i R , { i , i } ∩ { row ( j ) , row ( j ) } = ∅ , and since j , j C , { j , j } ∩ { col ( i ) , col ( i ) } = ∅ .Therefore, due to Lemma 5.2, { i , i } × { j , j } is a winning pair. Lemma 5.6 If n = 4 and (( | C | = 4 and | R | ≤ or ( | R | = 4 and | C | ≤ , then there exists awinning pair. Proof. Without loss of generality, assume that | C | = 4 and | R | ≤ 3. Now, we order the rows sothat R = { , , } , and we order the columns so that γ ii = ( b, a ) for all i ∈ [4], i.e., row ( j ) = j forall j ∈ [4]. If | R | = 2, then col (1) = 2 , col (2) = 1 and γ ij = ( b, b ) for i ∈ { , } , j ∈ { , } . So, { , } × { , } is a winning pair.Now let | R | = 3. Then, G = ( b, a ) · · ·· ( b, a ) · ·· · ( b, a ) · ( b, b ) ( b, b ) ( b, b ) ( b, a ) . We give special attention to the diagonal block D i = (cid:18) γ ii γ i ( i +1) γ ( i +1) i γ ( i +1)( i +1) (cid:19) for i ∈ [3] . γ ii = ( b, a ) for all i ∈ [4]. We say that D i is of type 1 if γ i ( i +1) = γ ( i +1) i = ( a, b ) andof type 2 if γ i ( i +1) = γ ( i +1) i = ( b, b ). Note that D i is a winning pair if and only if D i is of type 1or ( D i is of type 2 and row ( i ) = row ( i + 1)). We will construct all matrices whose 2 × (1) In the first step we construct all matrices without 2 × γ = ( a, b ) and γ i = ( b, b ) for i ∈ [2] and that the matrix isdetermined uniquely by γ . There are two cases: (1.1) γ = ( a, b ). Then ( b, a ) ( b, b ) ( a, b ) ( b, b )(a,b) ( b, a ) ( b, b ) (b,b)(b,b) ( a, b ) ( b, a ) (a,b)( b, b ) ( b, b ) ( b, b ) ( b, a ) and { , } × { , } is a winningpair. (1.2) γ = ( b, b ). Then ( b, a ) (a,b) ( b, b ) (b,b)( b, b ) ( b, a ) ( a, b ) ( b, b )( a, b ) (b,b) ( b, a ) (a,b)( b, b ) ( b, b ) ( b, b ) ( b, a ) and { , } × { , } is a winningpair. (2) In this step we allow in the construction of the matrices no blocks D i of type 1 and a blockof type 2 only if row ( i ) = row ( i + 1). Since | R | = 3, only one such i can exist. Furthermore,if row (3) = row (4), then row (4) = 3. There are four cases: (2.1) row (1) = row (2). Then, row (1) = row (2) = 3, which implies that row (4) = 3. Butthen | R | ≤ 2, a contradiction. (2.2) row (2) = row (3). Then, row (2) = row (3) = 1, which implies that row (4) = 3 and row (1) = 2. Hence, γ = γ = ( a, b ), and thus, D is of type 1. (2.3) row (3) = row (4) = 1. Then γ = ( b, b ) and this determines the matrix uniquely: ( b, a ) ( b, b ) ( a, b ) ( a, b )( a, b ) (b,a) ( b, b ) (b,b)( b, b ) ( a, b ) ( b, a ) ( b, b )( b, b ) (b,b) ( b, b ) (b,a) and { , } × { , } is a winning pair. (2.4) row (3) = row (4) = 2. Then γ = ( a, b ) and this determines the matrix uniquely: (b,a) ( a, b ) ( b, b ) (b,b)( b, b ) ( b, a ) ( a, b ) ( a, b )( a, b ) ( b, b ) ( b, a ) ( b, b )(b,b) ( b, b ) ( b, b ) (b,a) and { , } × { , } is a winning pair.So, up to reordering of rows and columns, there are 4 matrices for which no of the D i ’s is a winningpair. We have shown that these 4 matrices have a winning pair.We are now ready to prove the key result. 26 heorem 5.7 Let G be a 2-players, 2-values, n -strategies normal game with n ≥ and not n = | C | = | R | = 4 . Then a winning pair exists and can be found in time O ( n ) . Proof. Consider an algorithm that simulates the algorithms described in Lemmas 5.4 and 5.5. Sothis algorithm finds a winning pair for n ≥ | R | ≥ | C | ≥ 4, or |{ col ( i ) : i R }| ≥ |{ row ( j ) : j C }| ≥ 2. The case n = 4 and ( | C | = 4 and | R | ≤ | R | = 4 and | C | ≤ | R | ≤ | C | ≤ 3, and |{ col ( i ) : i R }| = 1.In the proof we use some appropriate numbering of rows and columns. We are aware thatthe algorithm would operate with the actual row and column numbers, but we feel that using therenumbering allows for an easier understanding of the algorithmic idea.By renumbering the columns we can obtain col ( i ) = 1 for i R . By renumbering the rows andthe columns 2 , . . . , n we can obtain additionally R = { , } if | R | = 2 and R = { , , } if | R | = 3,and row (1) = 1, row (2) = 2. Note that in this setting γ ( i, j ) = ( b, b ) for all i ≥ , j ≥ 2. Figure 2illustrates this initial setting. ❛❛❛❛❛❛❛ a, b ) ( a, b ) ( b, b ) ( b, a )( b, a ) 2 3 4 512345 R = { , } if | R | = 2 R = { , , } if | R | = 3 Figure 2: Illustrating the initial setting of the proof of Theorem 5.7.Since we do not need the information whether | R | = 2 or | R | = 3 in this first part of the proof,we make no assumptions about the entries in the third row at this stage of the proof.Next we discuss the influence of the values col (1) and col (2). If col (1) = 2 and col 2) = 1 or if col (1) = 2 and col (2) = 1 and col (1) = col (2), then { , } × { , } is a winning pair. In both cases row (1) = row (2) and col (1) = col (2). So, the following three cases remain:( i ) col (1) = 2 , col (2) = 1( ii ) col (2) = 1 , col (1) = 2( iii ) col (1) = col (2)Case ( ii ) is the more elaborate case. We start by discussing cases ( i ) and ( iii ). Cases ( i ) / ( iii ) Here, in both cases, γ (2 , 1) = ( b, b ) and col (2) = j ≥ γ (2 , j ) = ( b, a ). Then { , } × { , j } isa winning pair. This is true, since γ (4 , 1) = ( b, a ), γ (4 , j ) = ( b, b ), col (4) = 1 = j = col (2) and row (1) = 2 = row ( j ), since γ (2 , j ) = ( b, a ). This situation is illustrated in Figure 3.27 ❛❛❛❛❛❛ a, b )( b, b ) ( b, a )( a, b ) ( b, b ) ( b, a )( b, a ) 2 col (2)12345 Figure 3: Illustrating cases ( i ) / ( iii ) of the proof of Theorem 5.7. Case ( ii ) In this case γ (1 , 2) = ( b, b ) and γ (2 , 1) = ( b, a ). By renumbering the columns that are greater orequal to 3, we can obtain that col (1) = 3, i.e., γ (1 , 3) = ( b, a ). If row (3) = 2, then { , } × { , } isa winning pair. This true since in this case γ (1 , 1) = γ (2 , 3) = ( a, b ) and γ (1 , 3) = γ (2 , 1) = ( b, a ),and row (1) = 1 = 2 = row (2) and col (1) = 3 = 1 = col (1).So, there remain the case row (3) = 2. It is row (3) = 1 since γ (1 , 3) = ( b, a ). We are now in thecase | R | = 3 and row (3) = 3. This situation is illustrated in Figure 4. ❛❛❛❛❛❛❛ a, b ) ( b, b ) ( b, a )( b, a ) ( b, b )( a, b ) ( b, b ) ( b, a )( b, a ) ( a, b )2 3 4 512345 Figure 4: Illustrating case ( ii ) of the proof of Theorem 5.7.We now study the influence of col (3). If col (3) = 1, then { , } × { , } is a winning pair,since γ (1 , 1) = γ (3 , 3) = ( a, b ) and γ (1 , 3) = γ (3 , 1) = ( b, a ), and row (1) = 1 = 3 = row (3) and col (1) = 3 = 1 = col (3).Now let col (3) = 2, i.e., γ (3 , 1) = ( b, b ) and γ (3 , 2) = ( b, a ). Then { , } × { , } is a winningpair, since γ (3 , 1) = γ (4 , 2) = ( b, b ) and γ (3 , 3) = γ (4 , 1) = ( a, b ), and col (3) = 2 = 1 = col (4) and row (1) = 1 = 2 = row (2).Finally, consider the case col (3) ≥ 3. Since γ (3 , 3) = ( a, b ) this implies that col (3) ≥ γ (3 , 2) = ( b, b ) and { , } × { , } is a winning pair, since γ (2 , 2) = γ (3 , 3) = ( a, b ) and γ (2 , 3) = γ (3 , 2) = ( b, b ), and row (2)2 = 3 = row (3) and col (2) = 1 = col (3), completing proof ofthe theorem.Theorem 5.7, Lemma 5.1 and the definition of a winning pair (Definition 5.1) yield the followingresult: Corollary 5.8 Consider a 2-players, 2-values, n -strategies normal game G , and a unimodal valu-ation F with F (cid:0) (cid:1) = b . Then, G has an F -equilibrium that can be computed in O ( n ) time. Remark: Observe that Theorem 5.7 does not cover the cases n ≤ and n = | R | = | C | = 4 .However, we know from Theorem 5.9 (next section) that for n = 3 there always exists an F -equilibrium with F (cid:16) (cid:17) = b . From Corollary 3.5 we know that an F -equilibrium exists also for n = 2 . Regarding the case n = | R | = | C | = 4 , the strategy vector p i = p i = 1 / for i ∈ [ n ] is afully mixed equilibrium. Here we focus on two-players, two-values games with F (cid:0) (cid:1) ≤ b and 3 strategies (i.e., n = 3). In thiscase we have a complete picture on the existence and inexistence of F -equilibria (c.f. Corollary 5.10).We first show: Theorem 5.9 Consider a unimodal valuation F with F (cid:0) (cid:1) ≤ b . Then, every 2-players, 2-values,3-strategies game has an F -equilibrium. Proof. We start with an E -equilibrium ( p , p ) for G . If | σ ( p ) | = 1 or | σ ( p ) | = 1, then, byLemma 2.3, there is also a pure equilibrium for G . So assume that | σ ( p i ) | ≥ i ∈ [2]. By Lemma 2.4, assume, without loss of generality, that if for a player i ∈ [2], | σ ( p i ) | = 2,then p i ( j ) = 12 for each strategy j ∈ σ ( p i ). If | σ ( p ) | = | σ ( p ) | = 3, then ( p , p ) is a fully mixed E -equilibrium; hence, by Corollary 3.4, it is also an F -equilibrium. So assume, without loss ofgenerality, that | σ ( p ) | = 2, with σ ( p ) = { , } . We distinguish two cases with respect to | σ ( p ) | : (A) | σ ( p ) | = 2, with σ ( p ) = { , } : The idea of the proof is to show that ( p , p ) is also an F -equilibrium. Since player 1 is E -constant on σ ( p ), Lemma 3.2 implies that she is also F -constant on σ ( p ). So it remains to prove that player 1 cannot F -improve by switching tostrategy 3. By assumption, p (1) = p (2) = 12 . We distinguish again the three cases from theproof of Lemma 2.4: (A/1) µ ( k, 1) = µ ( k, 2) = a for all k ∈ σ ( p ): Then, x ( p , p ) = 1, so that V ( p , p ) = F (1) = a . Thus, player 1 cannot F -improve. (A/2) µ ( k, 1) = µ ( k, 2) = b for all k ∈ σ ( p ): Then, x ( p , p ) = 0, so that V ( p , p ) = F (0) = b . Since ( p , p ) is an E -equilibrium, it must also hold that µ (3 , 1) = µ (3 , 2) = b .Thus, x ( p , p ) = 0, so that V ( p , p ) = F (0) = b , and player 1 cannot F -improve. (A/3) µ ( k, = µ ( k, 2) for all k ∈ σ ( p ): Then, x ( p , p ) = 12 , so that V ( p , p ) = F (cid:16) (cid:17) .Since player 1 is E -happy, there are only two cases (the case µ (3 , 1) = µ (3 , 2) = a isexcluded since in this case player 1 is not E -happy):29 A/3/i) µ (3 , = µ (3 , x ( p , p ) = 12 , so that V ( p , p ) = F (cid:16) (cid:17) . So, player 1cannot F -improve. (A/3/ii) µ (3 , 1) = µ (3 , 2) = b : Then, x ( p , p ) = 0, so that V ( p , p ) = F (0) = b . Since F (cid:16) (cid:17) ≤ b , player 1 cannot F -improve.So player 1 cannot F -improve by switching to strategy 3. Due to the symmetry between thetwo players, player 2 cannot improve either by switching to strategy 3. (B) | σ ( p ) | = 3: Then, Corollary 3.4 implies that player 1 is F -happy. The idea of the proof is toshow that either ( p , p ) is also an F -equilibrium, or define a new probability distribution forplayer 1 so that the resulting mixed profile is an F -equilibrium. The reason we can do this is thefollowing: Because of the Optimal-Value Property (c.f. Definition 2.2) player 1 is V -constanton σ ( p ); this together with Lemma 3.2 imply that V ( p , p ) = F ( x ( p , p )) = F ( x ( p s i , p ))for all s i ∈ σ ( p ). Now define a new mixed strategy b p with σ ( b p ) ⊆ σ ( p ). Then observethat player 1 remains F -happy.So, what it remains is to consider the F -happiness of player 2. By the WEEP for player 2, E ( p , p ) = E ( p , p ). We use the following observation:Since F (cid:18) (cid:19) ≤ b, F ( y ) ≥ F ( z ) , for all y < z, z ≥ / . (1)Now, assume, without loss of generality, that µ (1 , ≤ µ (1 , ≤ µ (1 , (B/1) µ ( k, 1) = µ ( k, 2) for all strategies k ∈ [3]: There are three subcases: (B/1/i) µ ( k, 1) = a for all strategies k ∈ [3]: Then, x ( p , p ) = 1, so that V ( p , p ) = F (1) = a . So, player 2 cannot F -improve. (B/1/ii) µ ( k, 1) = b for all strategies k ∈ [3]: Then, since σ ( p ) = { , } , E ( p , p ) = b . Byassumption, ( p , p ) is an E -equilibrium. This implies that µ ( k, 3) = b for allstrategies k ∈ [3] and V ( p , p ) = V ( p , p ) = F (0) = b . So, player 2 cannot F -improve. (B/1/iii) µ (1 , 1) = a and µ (3 , 1) = b : Since player 2 is E -happy with ( p , p ), E ( p , p ) ≤ E ( p , p ). There are two subcases to consider: • µ (2 , 1) = a : Then, µ = a a ∗ a a ∗ b b ∗ , wherean ∗ is an arbitrary value from { a, b } .By assumption, ( p , p ) is an E -Equilibrium. This implies that there issome i ∈ [3] with µ ( i, 3) = b . Define a new probability distribution b p for player1 by b p ( k ) = 1 / 3, for all k ∈ [3] . Then, V ( b p , p ) = F (cid:16) (cid:17) and V ( b p , p ) = F ( x )with some x ≤ / 3. By observation (1), F ( x ) ≥ F (cid:16) (cid:17) . Hence, ( b p , p ) is an F -equilibrium, since player 2 cannot F -improve by switching to strategy 3.30 µ (2 , 1) = b : Then, µ = a a ∗ b b ∗ b b ∗ . If µ (1 , 3) = a , then µ (2 , 3) = µ (3 , 3) = b , since otherwise player 2 could E -improve by choosing strategy3, contradicting the assumption that ( p , p ) is an E -Equilibrium.So, consider now the case that µ (1 , 3) = b . Define a new probability distribution b p for player 1 by b p (1) = 1 / 2, and b p (2) = b p (3) = 1 / 4. Then, V ( b p , p ) = F (cid:16) (cid:17) and V ( b p , p ) = F ( x ) with some x ≤ / 2. By observation (1), F ( x ) ≥ F (cid:16) (cid:17) . Hence, ( b p , p ) is an F -equilibrium, since player 2 cannot F -improve byswitching to strategy 3. (B/2) There is a strategy b k ∈ [3] with µ ( b k, = µ ( b k, σ ( p ) dominates some other strategy in σ ( p ) with respect to σ ( p ).Hence, there is at least one other e k ∈ [3] with µ ( e k, = µ ( e k, WEEP forplayer 2, E ( p , p ) = E ( p , p ). We distinguish four cases, each represented by thematrix ( µ ( k, j )) ≤ k,j ≤ , where, as before, ∗ is an arbitrary value from { a, b } : (B/2/i) µ = a a ∗ a b ∗ b a ∗ : By assumption, ( p , p ) is an E -Equilibrium. This impliesthat there is some i ∈ [3] with µ ( i, 3) = b . Define a new probability distribution b p for player 1 by b p ( k ) = 1 / 3, for all k ∈ [3] . Then, V ( b p , p ) = F (cid:16) (cid:17) and V ( b p , p ) = F ( x ) with some x ≤ / 3. As in case (B/1/iii) , and by observation (1),we conclude that player 2 cannot F -improve by switching to strategy 3. (B/2/ii,iii) We consider two subcases: • µ = a b ∗ a b ∗ b a ∗ : The WEEP for player 2 implies p (1)+ p (2) = p (3) = 1 / V ( p , p ) = F (cid:16) (cid:17) . If µ (3 , 3) = a , then ( p , p ) being an E -equilibriumimplies that µ (1 , 3) = µ (2 , 3) = b . Thus, ( p , p ) is also an F -equilibrium.If µ (3 , 3) = b , then V ( p , p ) = F ( x ) with some x ≤ / F -improve by switching to strategy 3. • µ = a b ∗ b a ∗ b a ∗ : This case is equivalent to the above case, by interchangingthe first and second columns and the first and third rows. (B/2/iv) µ = a b ∗ b a ∗ b b ∗ : The WEEP for player 2 implies p (1) = p (2). Since ( p , p ) isan E -equilibrium, then µ ( i, 3) = a for at most one value of i ∈ [3]. Define a newprobability distribution b p for player 1. We distinguish two cases: • If µ ( i, 3) = a for exactly one value i ∈ [3] or if F (cid:16) (cid:17) ≤ b , then b p = 1 / k ∈ [3]. Then V ( b p , p ) = F (cid:16) (cid:17) and player 2 cannot F -improve by choosingstrategy 3. 31 If µ ( i, 3) = b for all i ∈ [3] and F (cid:16) (cid:17) > b , then set σ ( b p ) = { , } and b p (1) = b p (2) = 1 / 2. Player 1 cannot F -improve since V ( p i , p ) has the samevalue for all i ∈ [3], and player 2 cannot F -improve since V ( b p , p ) = F (cid:16) (cid:17) ≤ b and V ( b p , p ) = F (0) = b .The claim follows.Now observe that by Theorem 4.3 we conclude that the 2-players, 2-values and 3-strategies normalgame with bimatrix C = ( a, b ) ( b, a ) ( a, b )( b, a ) ( a, b ) ( b, b )( b, b ) ( b, b ) ( b, a ) , has no F -equilibrium when F (cid:0) (cid:1) > b . This complements the result of Theorem 5.9. In conclusion,for 2-players, 2-values games with 3 strategies we have: Corollary 5.10 For a unimodal valuation F the following properties hold: ( i ) If F (cid:0) (cid:1) ≤ b , then every 2-player, 2 values, 3-strategies game has an F -equilibrium. ( ii ) If F (cid:0) (cid:1) > b , then there exists a normal 2-player, 2 values, 3-strategies game without F -equilibrium. In this work we have investigated the (in)existence of equilibria for 2-players, 2-values games underunimodal valuations. Our work is the first to adopt the combination of abstract settings of unimodalvaluations and 2-values sparse bimatrix games, such as normal games. Normal games are a class ofcounterexample games, which provide a canonical way of the sparsest bimatrix games. Unimodalvaluations provide the simplest possible way of specifying a non-monotone valuation function.Combining these two critical parameters enables the simplest modeling of sparsity and payoffconcavity.The most striking open problem arising from the context of this paper is the special role ofthe condition F ( ) = b , documented for normal games by Theorem 4.4 and Corollary 5.8. Will F ( ) = b guarantee the existence of F -equilibria also under relaxed normality, e.g., by allowing two a ’s per row and column, respectively? Winning pairs are not necessarily F -equilibria in this case.Another open problem is the complexity of computing F -equilibria for 2-values games. Verylikely the problem is N P -hard. 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