An Existence Theorem of Nash Equilibrium in Coq and Isabelle

Roux, Stéphane; Martin-Dorel, Érik; Smaus, Jan-Georg

doi:10.4204/eptcs.256.4

Cited by 5 publications

(6 citation statements)

References 13 publications

(31 reference statements)

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“…Then implication (S3) ⇒ (S1) was proven in [16]. The same proof was repeated in a more focused paper [17]; see also [28].…”

Section: Theorem 1 ([16]mentioning

confidence: 79%

Backward induction in presence of cycles

Gurvich

2018

Journal of Logic and Computation

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For the classical backward induction algorithm, the input is an arbitrary n-person positional game with perfect information modeled by a finite acyclic directed graph (digraph) and the output is a profile (x1, . . . , xn) of pure positional strategies that form some special subgame perfect Nash equilibrium. We extend this algorithm to work with digraphs that may have directed cycles. Each digraph admits a unique partition into strongly connected components, which will be treated as the outcomes of the game. Such a game will be called a deterministic graphical multistage(DGMS) game. If we identify the outcomes corresponding to all strongly connected components, except terminal positions, we obtain the so-called deterministic graphical(DG) games, which are frequent in the literature. The outcomes of a DG game are all terminal positions and one special outcome c that is assigned to all infinite plays. We modify the backward induction procedure to adapt it for the DGMS games. However, by doing so, we lose two important properties: the modified algorithm always outputs a Nash equilibrium (NE) only when n = 2 and, even in this case, this NE may be not subgame perfect.(Yet, in the zero-sum case it is.) The lack of these two properties is not a fault of the algorithm, just (subgame perfect) Nash equilibria in pure positional strategies may fail to exist in the considered game.

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“…Then implication (S3) ⇒ (S1) was proven in [16]. The same proof was repeated in a more focused paper [17]; see also [28].…”

Section: Theorem 1 ([16]mentioning

confidence: 79%

Backward induction in presence of cycles

Gurvich

2018

Journal of Logic and Computation

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“…We will apply some general criteria of Nash-solvability (based on the concept of Boolean duality and developed for the two-person game forms and correspondences in [13], [15], [27]) to the monotone bargaining case considered in this paper. These criteria imply that G = G m,n is NS for all m and n. Moreover, given m and n, the same NE (x, y) exists for all g ∈ G, and this equilibrium is simple, that is, G(x, y) contains a unique deal.…”

Section: Introduction 1main Resultsmentioning

confidence: 99%

“…Moreover, they have the same pair of dual hypergraphs (or the same self-dual effectivity functions, in terminology of [16] and [17]). It is known that tightness and Nash-solvability are equivalent properties of the two-person game forms; see [13], [15], and also [27]. In Section 3.3 we recall a constructive proof of this fact that is based on an algorithm finding a NE in any game (g, u) in time linear in the size of the set O of its outcomes, whenever g is tight.…”

Section: Introduction 1main Resultsmentioning

confidence: 99%

Monotone bargaining is Nash-solvable

Gurvich

Koshevoy²

2018

Discrete Applied Mathematics

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Given two finite ordered sets A = {a 1 , . . . , a m } and B = {b 1 , . . . , b n }, introduce the set of mn outcomes. . , n}. Two players, Alice and Bob, have the sets of strategies X and Y that consist of all monotone non-decreasing mappings x : A → B and y : B → A, respectively. It is easily seen that each pair (Choose an arbitrary deal g(x, y) ∈ G(x, y) to obtain a mapping g : X × Y → O, which is a game form. We use notation g ∈ G and show that each such game form is tight and, hence, Nash-solvable, that is, for any pair u = (u A , u B ) of utility functions u A : O → R of Alice and u B : O → R of Bob, the obtained monotone bargaining game (g, u) has at least one Nash equilibrium in pure strategies. Moreover, the same equilibrium can be selected for all g ∈ G. We also obtain an efficient algorithm that determines such an equilibrium in time linear in mn, although the numbers of strategies |X| = m+n−1 m and |Y | = m+n−1 n are exponential in mn. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.

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“…Notice other works using proof assistants for proving properties of agents. For instance, Stéphane Le Roux proved the existence of Nash equilibria using Coq and Isabelle [10,11]. In a somewhat connected area, Tobias Nipkow proved Arrows theorem in HOL [8].…”

Section: Introductionmentioning

confidence: 99%