Equilibria in bottleneck games

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi (1973a), see also Rubinstein (1991), and Aumann and Brandenburger (1991), a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others.

A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than can be obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qubits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.

We investigate learning in a repeated bank choice game, where agents first choose a bank to deposit in and then decide to withdraw that deposit or not. This game has a single Nash equilibrium in pure strategies, characterized by all agents depositing in the bank that offers the highest return, even though it may be more vulnerable to bankruptcy if some agents withdraw early. We use an individual evolutionary learning algorithm to model under which circumstances and with which beliefs agents can learn the Nash equilibrium in the repeated game and compare the results to an experiment. We find participants coordinating on the Nash equilibrium in the presence of low or medium vulnerability banks, but efficient coordination fails when both banks are highly vulnerable (irrespective of whether participants have full or only partial information).

Multiplicity, Instability and Sunspots in Games

A consignment contract with revenue sharing between an app developer and a distribution platform

In this paper we deal with linear production situations in which there is a limited common-pool resource, managed by an external agent. The profit that a producer, or a group of producers, can attain depends on the amount of common-pool resource obtained through a certain procedure. We contemplate a competitive process among the producers or groups of producers and study the corresponding non cooperative games, describing their (strict) Nash equilibria in pure strategies. It is shown that strict Nash equilibria form a subset of strong Nash equilibria, which in turn form a proper subset of Nash equilibria.

The determination of wage rates and skill requirements for jobs is studied in a model where workers, who vary in ability, acquire skill through (costly) training. If firms' skill requirements are fixed, there exists a unique Nash equilibrium in wage rates, but the resulting allocation of labor is inefficient. The Nash equilibrium wage rates and allocation of labor approach the competitive equilibrium if and only if the number of job types increases on every skill range and no firm remains a monopsonist on any skill range. If firms choose both skill requirements and wage rates, no Nash equilibrium in pure strategies exists.

A pure-strategy equilibrium existence theorem is extended to include games with non-expected utility players. It is shown that to guarantee the existence of a Nash equilibrium in pure strategies, the linearity of preferences in the probabilities can be replaced by the weaker requirement of quasiconvexity in the probabilities.

We propose a new fixed point theorem that completely characterizes the existence of fixed points for multivalued maps on finite sets. Our result can be seen as a generalization of Abian’s fixed point theorem. In the context of finite games, our result can be used to characterize the existence of a Nash equilibrium in pure strategies and can therefore distinguish pure strategy equilibria from mixed strategy equilibria in the celebrated Nash theorem.

Non-Zero-Sum Stopping Games in Discrete Time

A full game theoretic analysis of the two-person game with incomplete information is not available in the literature. It might be helpful to know something about the set of the equilibrium points of this game in order to make a prediction of the experimental results and the shape of the strategies. Therefore, we shall solve the game for Nash equilibria in pure strategies. A refinement of the equilibrium concept, to sequential equilibria for example, would be a narrowing from a behavioral point of view. It is not obvious that subjects follow the concept of sequential rationality. Since a large variety of possible behavior in equilibrium should be captured by the analysis we shall concentrate on the set of pure Nash equilibria. The restriction to pure strategies was chosen in order to simplify the computations. In this respect it is of interest that in the strategy experiment only pure strategies are observed.

There are many situations in which a customer's proclivity to buy the product of any firm depends not only on the classical attributes of the product such as its price and quality, but also on who else is buying the same product. We model these situations as games in which firms compete for customers located in a “social network”. Nash Equilibrium (NE) in pure strategies exist and are unique. Indeed there are closed-form formulae for the NE in terms of the exogenous parameters of the model, which enables us to compute NE in polynomial time. An important structural feature of NE is that, if there are no a priori biases between customers and firms, then there is a cut-off level above which high cost firms are blockaded at an NE, while the rest compete uniformly throughout the network. We finally explore the relation between the connectivity of a customer and the money firms spend on him. This relation becomes particularly transparent when externalities are dominant: NE can be characterized in terms of the invariant measures on the recurrent classes of the Markov chain underlying the social network.

Dynamic spectrum management (DSM) has been proposed to achieve next-generation rates on digital subscriber lines (DSL). Because the copper twisted-pair plant is an interference-constrained environment, the multiuser performance and spectral compatibility of DSM schemes are of primary concern in such systems. While the analysis of multiuser interference has been standardized for current static spectrum-management (SSM) techniques, at present no corresponding standard DSM analysis has been established. This paper examines a multiuser spectrum-allocation problem and formulates a lower bound to the achievable rate of a DSL modem that is tight in the presence of the worst-case interference. A game-theoretic analysis shows that the rate-maximizing strategy under the worst-case interference (WCI) in the DSM setting corresponds to a Nash equilibrium in pure strategies of a certain strictly competitive game. A Nash equilibrium is shown to exist under very mild conditions, and the rate-adaptive waterfilling algorithm is demonstrated to give the optimal strategy in response to the WCI under a frequency-division (FDM) condition. Numerical results are presented for two important scenarios: an upstream VDSL deployment exhibiting the near-far effect, and an ADSL RT deployment with long CO lines. The results show that the performance improvement of DSM over SSM techniques in these channels can be preserved by appropriate distributed power control, even in worst-case interference environments.

On Nash-solvability in pure stationary strategies of finite games with perfect information which may have cycles

On extremal pure Nash equilibria for mixed extensions of normal-form games