Nash equilibrium strategies in discrete-time finite-horizon dynamic games with risk-and effort-averse players

The objective of this paper is to re-examine the risk-and effort attitude in the context of strategic dynamic interactions stated as a discrete-time finite-horizon Nash game. The analysis is based on the assumption that players are endogenously risk-and effort-averse. Each player is characterized by distinct risk-and effort-aversion types that are unknown to his opponent. The goal of the game is the optimal risk-and effort-sharing between the players. It generally depends on the individual strategies adopted and, implicitly, on the the players' types or characteristics.

Invariance and randomness in the Nash program for coalitional games

By introducing physical outcomes in coalitional games we note that coalitional games and social choice problems are equivalent (implying that so are the theory of implementation and the Nash program). This facilitates the understanding of the role of invariance and randomness in the Nash program. Also, the extent to which mechanisms in the Nash program perform ``real implementation'' is examined.

Existence of nash equilibrium in competitive nonlinear pricing games with adverse selection

We show that for a large class of competitive nonlinear pricing games with adverse selection, the property of better-reply security is naturally satisfied - thus, resolving via a result due to Reny (1999) the issue of existence of Nash equilibrium for a large class of competitive nonlinear pricing games.

Spike-based decision learning of nash equilibria in two-player games

Humans and animals face decision tasks in an uncertain multi-agent environment where an agent's strategy may change in time due to the co-adaptation of others strategies. The neuronal substrate and the computational algorithms underlying such adaptive decision making, however, is largely unknown. We propose a population coding model of spiking neurons with a policy gradient procedure that successfully acquires optimal strategies for classical game-theoretical tasks. The suggested population reinforcement learning reproduces data from human behavioral experiments for the blackjack and the inspector game. It performs optimally according to a pure (deterministic) and mixed (stochastic) Nash equilibrium, respectively. In contrast, temporal-difference(TD)-learning, covariance-learning, and basic reinforcement learning fail to perform optimally for the stochastic strategy. Spike-based population reinforcement learning, shown to follow the stochastic reward gradient, is therefore a viable candidate to explain automated decision learning of a Nash equilibrium in two-player games.

On the implementation of the L-Nash bargaining solution in two-person bargaining games

The “Nash program” initiated by Nash (Econometrica 21:128–140, 1953) is a research agenda aiming at representing every axiomatically determined cooperative solution to a game as a Nash outcome of a reasonable noncooperative bargaining game. The L-Nash solution first defined by Forgó (Interactive Decisions. Lecture Notes in Economics and Mathematical Systems, vol 229. Springer, Berlin, pp 1–15, 1983) is obtained as the limiting point of the Nash bargaining solution when the disagreement point goes to negative infinity in a fixed direction. In Forgó and Szidarovszky (Eur J Oper Res 147:108–116, 2003), the L-Nash solution was related to the solution of multiciteria decision making and two different axiomatizations of the L-Nash solution were also given in this context. In this paper, finite bounds are established for the penalty of disagreement in certain special two-person bargaining problems, making it possible to apply all the implementation models designed for Nash bargaining problems with a finite disagreement point to obtain the L-Nash solution as well. For another set of problems where this method does not work, a version of Rubinstein’s alternative offer game (Econometrica 50:97–109, 1982) is shown to asymptotically implement the L-Nash solution. If penalty is internalized as a decision variable of one of the players, then a modification of Howard’s game (J Econ Theory 56:142–159, 1992) also implements the L-Nash solution.

The game to play: expanding the co-opetition proposal through the strategic games matrix

Purpose - Using Brandenburger and Nalebuff`s 1995 co-opetition model as a reference, the purpose of this paper is to seek to develop a tool that, based on the tenets of classical game theory, would enable scholars and managers to identify which games may be played in response to the different conflict of interest situations faced by companies in their business environments. Design/methodology/approach - The literature on game theory and business strategy are reviewed and a conceptual model, the strategic games matrix (SGM), is developed. Two novel games are described and modeled. Findings - The co-opetition model is not sufficient to realistically represent most of the conflict of interest situations faced by companies. It seeks to address this problem through development of the SGM, which expands upon Brandenburger and Nalebuff`s model by providing a broader perspective, through incorporation of an additional dimension (power ratio between players) and three novel, respectively, (rival, individualistic, and associative). Practical implications - This proposed model, based on the concepts of game theory, should be used to train decision- and policy-makers to better understand, interpret and formulate conflict management strategies. Originality/value - A practical and original tool to use game models in conflict of interest situations is generated. Basic classical games, such as Nash, Stackelberg, Pareto, and Minimax, are mapped on the SGM to suggest in which situations they Could be useful. Two innovative games are described to fit four different types of conflict situations that so far have no corresponding game in the literature. A test application of the SGM to a classic Intel Corporation strategic management case, in the complex personal computer industry, shows that the proposed method is able to describe, to interpret, to analyze, and to prescribe optimal competitive and/or cooperative strategies for each conflict of interest situation.

The stability of the equilibrium outcomes in the admission games induced by stable matching rules

A stable matching rule is used as the outcome function for the Admission game where colleges behave straightforwardly and the students` strategies are given by their preferences over the colleges. We show that the college-optimal stable matching rule implements the set of stable matchings via the Nash equilibrium (NE) concept. For any other stable matching rule the strategic behavior of the students may lead to outcomes that are not stable under the true preferences. We then introduce uncertainty about the matching selected and prove that the natural solution concept is that of NE in the strong sense. A general result shows that the random stable matching rule, as well as any stable matching rule, implements the set of stable matchings via NE in the strong sense. Precise answers are given to the strategic questions raised.

An existence result for discontinuous games

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin (1986) and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou (1993) and Reny (1999). Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity.

On the existence of pure-strategy equilibria in large games

Over the years, several formalizations and existence results for games with a continuum of players have been given. These include those of Schmeidler (1973), Rashid (1983), Mas-Colell (1984), Khan and Sun (1999) and Podczeck (2007a). The level of generality of each of these existence results is typically regarded as a criterion to evaluate how appropriate is the corresponding formalization of large games. In contrast, we argue that such evaluation is pointless. In fact, we show that, in a precise sense, all the above existence results are equivalent. Thus, all of them are equally strong and therefore cannot rank the different formalizations of large games.

Implementation of the Walrasian correspondence by market games

In this paper we present a set of axioms guaranteeing that, in exchange economies with or without indivisible goods, the set of Nash, Strong and active Walrasian Equilibria all coincide in the framework of market games.

Equilibrium play and best response to (Stated) beliefs in constant sum games

We report experimental results on one-shot two person 3x3 constant sum games played by non-economists without previous experience in the laboratory. Although strategically our games are very similar to previous experiments in which game theory predictions fail dramatically, 80% of actions taken in our experiment coincided with the prediction of the unique Nash equilibrium in pure strategies and 73% of actions were best responses to elicited beliefs. We argue how social preferences, presentation effects and belief elicitation procedures may influence how subjects play in simple but non trivial games and explain the diferences we observe with respect to previous work.

A class of stochastic games with infinitely many interacting agents related to Glauber dynamics on random graphs

We introduce and study a class of infinite-horizon nonzero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove “fixation”, i.e. that players will adopt a constant strategy after a finite time. The resulting dynamics is related to zerotemperature Glauber dynamics on random graphs of possibly infinite volume.

Testing the TASP: An Experimental Investigation of Learning in Games with Unstable Equilibria

We report experiments designed to test between Nash equilibria that are stable and unstable under learning. The “TASP” (Time Average of the Shapley Polygon) gives a precise prediction about what happens when there is divergence from equilibrium under fictitious play like learning processes. We use two 4 x 4 games each with a unique mixed Nash equilibrium; one is stable and one is unstable under learning. Both games are versions of Rock-Paper-Scissors with the addition of a fourth strategy, Dumb. Nash equilibrium places a weight of 1/2 on Dumb in both games, but the TASP places no weight on Dumb when the equilibrium is unstable. We also vary the level of monetary payoffs with higher payoffs predicted to increase instability. We find that the high payoff unstable treatment differs from the others. Frequency of Dumb is lower and play is further from Nash than in the other treatments. That is, we find support for the comparative statics prediction of learning theory, although the frequency of Dumb is substantially greater than zero in the unstable treatments.

Dominance Solvable Games with Multiple Payoff Criteria.

Two logically distinct and permissive extensions of iterative weak dominance are introduced for games with possibly vector-valued payoffs. The first, iterative partial dominance, builds on an easy-to check condition but may lead to solutions that do not include any (generalized) Nash equilibria. However, the second and intuitively more demanding extension, iterative essential dominance, is shown to be an equilibrium refinement. The latter result includes Moulin’s (1979) classic theorem as a special case when all players’ payoffs are real-valued. Therefore, essential dominance solvability can be a useful solution concept for making sharper predictions in multicriteria games that feature a plethora of equilibria.

Simultaneous Nash bargaining with consistent beliefs

We propose and analyze a new solution concept, the R solution, for three-person, transferable utility, cooperative games. In the spirit of the Nash Bargaining Solution, our concept is founded on the predicted outcomes of simultaneous, two-party negotiations that would be the alternative to the grand coalition. These possibly probabilistic predictions are based on consistent beliefs. We analyze the properties of the R solution and compare it with the Shapley value and other concepts. The R solution exists and is unique. It belongs to the bargaining set and to the core whenever the latter is not empty. In fact, when the grand coalition can simply execute one of the three possible bilateral trades, the R solution is the most egalitarian selection of the bargaining set. Finally, we discuss how the R solution changes important conclusions of several well known Industrial Organization models.