On equilibria in finite games

We are concerned with Nash equilibrium points forn-person games. It is proved that, given any real algebraic numberα, there exists a 3-person game with rational data which has a unique equilibrium point andα is the equilibrium payoff for some player. We also present a method which allows us to reduce an arbitraryn-person game to a 3-person one, so that a number of questions about generaln-person games can be reduced to consideration of the special 3-person case. Finally, a completely mixed game, where the equilibrium set is a manifold of dimension one, is constructed.     

Pure Nash equilibria in finite two-person non-zero-sum games

In this paper we study bimatrix games. The payoff matrices have properties closely related to concavity of functions. For such games we find sufficient conditions for the existence of pure Nash equilibria.     

Pure strategy equilibria in symmetric two-player zero-sum games

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.     

Nonrandomized strategy equilibria in noncooperative stochastic games with additive transition and reward structure

In this paper, we study a discounted noncooperative stochastic game with an abstract measurable state space, compact metric action spaces of players, and additive transition and reward structure in the sense of Himmelberget al. (Ref. 1) and Parthasarathy (Ref. 2). We also assume that the transition law of the game is absolutely continuous with respect to some probability distributionp of the initial state and together with the reward functions of players satisfies certain continuity conditions. We prove that such a game has an equilibrium stationary point, which extends a result of Parthasarathy from Ref. 2, where the action spaces of players are assumed to be finite sets. Moreover, we show that our game has a nonrandomized (- )-equilibrium stationary point for each >0, provided that the probability distributionp is nonatomic. The latter result is a new existence theorem.     

Pure strategy equilibria in a class of systems defense games

This paper analyzes a class of systems defense games. Section 1 provides an overview. Section 2 reviews the assumptions used to specify a class of system defense games in Shubik and Weber (1978, 1981) and Shubik (1982). This is followed by a review of the additive games considered in Shubik and Weber (1981) — and the notion of comparable amounts of strategic resources used in that paper. Section 3 generalizes two of the features of additive system defense games with comparable resources. Section 4 states the final assumptions that are being made, and discusses the nature of the resulting class of systems defense games. The central result in the paper is in Section 5. The theorem specifically identifies an implicit objective function that is maximized by the equilibrium strategies. In Section 6, the theorem is used to also obtain an equilibrium existence result and identify conditions where there is a unique equilibrium.I gratefully acknowledge financial support provided by a Pew Grant to the University of Maryland for research on Economics and National Security. I also gratefully acknowledge helpful comments and suggestions provided by the participants in Maryland's Economics and National Security Luncheon Seminar and an anonymous referee.     

Some properties for the intersection of Moran sets with their translates

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given.     

Pure strategy Nash equilibria in non-zero sum colonel Blotto games

We analyze a Colonel Blotto game in which opposing parties have differing relative intensities. In other words, the game is non-zero sum because colonels have asymmetric and heterogeneous battlefield valuations. We characterize the colonels’ payoffs that sustain a pure strategy equilibrium and present an algorithm that reaches the equilibrium actions (when they exist). Finally we show that the set of games with a pure strategy equilibria is non-empty.     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.     

Limiting distributions of the number of pure strategy Nash equilibria in n-person games

We study the number of pure strategy Nash equilibria in a “random” n-person non-cooperative game in which all players have a countable number of strategies. We consider both the cases where all players have strictly and weakly ordinal preferences over their outcomes. For both cases, we show that the distribution of the number of pure strategy Nash equilibria approaches the Poisson distribution with mean 1 as the numbers of strategies of two or more players go to infinity. We also find, for each case, the distribution of the number of pure strategy Nash equilibria when the number of strategies of one player goes to infinity, while those of the other players remain finite.     

A test of the core solution in finite strategy non-sidepayment games

This paper reports a test of the core solution in cooperative non-sidepayment games where players have finite strategy sets. Two laboratory experiments were conducted with three-person and four-person games; in both experiments, the core solution was tested competitively against the von Neumann-Morgenstern stable set and the imputation set. Predictions from these solution concepts were computed under parameters of α-effectiveness and strict preference. Results show that the frequency of outcomes falling in core is substantially higher than that observed in previous experiments (most of which involve sidepayment games). In addition, goodness-of-fit tests show that the core solution predicts the observed outcomes more accurately than do the stable set or the imputation set.     

Some properties of Baire sets and sets of positive measure

In un lavoro classico Steinhaus ha dimostrato undici teoremi interessanti nei quali si considerano insiemi distanziali di sottoinsiemi della linea reale,aventi misura di Lebesgue positiva. Majumder ha dimostrato che sono validi teoremi analoghi a quelli di Steinhaus per insiemi del tipoR(E)={x/y: x,y∈E} doveE∪R; 0?E. In questo lavoro si considerano funzioni generali e si investiga sui quali degli undici teoremi di Steinhaus possono essere generalizzati. Nell'articolo, inoltre, usando una funzione generalef, vengono dati risultati analoghi a quelli di Steinhaus per insiemi di Baire.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

Antipodality properties of finite sets in Euclidean space

This is a survey of known results and still open problems on antipodal properties of finite sets in Euclidean space. The exposition follows historical lines and takes into consideration both metric and affine aspects.     

Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games

Recently, the concept of classical bargaining set given by Aumann and Maschler in 1964 has been extended to fuzzy bargaining set. In this paper, we give a modification to correct some weakness of this extension. We also extend the concept of the Mas-Colell's bargaining set (the other major type of bargaining sets) to its corresponding fuzzy bargaining set. Our main effort is to prove existence theorems for these two types of fuzzy bargaining sets. We will also give necessary and sufficient conditions for these bargaining sets to coincide with the Aubin Core in a continuous superadditive cooperative fuzzy game which has a crisp maximal coalition of maximum excess at each payoff vector. We show that both Aumann-Maschler and Mas-Colell fuzzy bargaining sets of a continuous convex cooperative fuzzy game coincide with its Aubin core.     

Some weak covering properties and infinite games

We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game G_fin(O,D_o). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz spaces, Topology Appl., 1995, 64(1), 9–21], too. The selection principle S₁(D_o,D_o) is also discussed in view of a special base. We show that every subspace of a hereditarily Lindelöf space with an ortho-base satisfies S₁(D_o,D_o).     

A relation between perfect equilibria in extensive form games and proper equilibria in normal form games

The concept of quasi-perfect equilibria for games in extensive form is introduced. It is shown that a proper equilibrium of a normal form game induces a quasi-perfect equilibrium in every extensive form game having this normal form.     

On equilibria in repeated games with absorbing states

We prove the existence of ε-(Nash) equilibria in two-person non-zerosum limiting average repeated games with absorbing states. These are stochastic games in which all states but one are absorbing. A state is absorbing if the probability of ever leaving that state is zero for all available pairs of actions.     

On systems of finite sets with constraints on their unions and intersections

Let F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ? n ? r, and |A ∩ B| ? s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ? f(n ? 1, r ? 1, s) + f(n ? 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1.