In this paper we explore the impact of imitation rules on players' long-run behaviors in evolutionary prisoner's dilemma games. All players sit sequentially and equally spaced around a circle. Players are assumed to interact only with their neighbors, and to imitate either their successful neighbors and/or themselves or the successful actions taken by their neighbors and/or themselves. In the imitating-successful-player dynamics, full defection is the unique long-run equilibrium as the probability of players' experimentations (or mutations) tend to 0. By contrast, full cooperation could emerge in the long run under the imitating-successful-action dynamics. Moreover, it is discovered that the convergence rate to equilibrium under local interaction could be slower than that under global interaction.

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of l (0 < l < ½), so that the gambler can stake no more than l, Wilkins (1972) showed that bold play remains optimal provided that 1 / l is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < l < ½ and all 0 < w < ½, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational l with 1 / l not an integer and all 0 < w < ½, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.

We consider stochastic replicator processes for games that are composed of finitely many trials. Several general results on the relation between Nash equilibria and the long-run behaviour of the stochastic processes are proved. In particular, a sufficient condition is given for almost sure convergence to a state where everyone plays in every trial a strict Nash equilibrium. The results are applied to multiple-trial conflicts based on wars of attrition and on sperm competition games with fair raffles, respectively.

In this paper we analyze players' long-run behavior in evolutionary coordination games with imperfect monitoring in a large population. Players can observe signals corresponding to other players' unseen actions and use the proposed simple or maximum likelihood estimation algorithm to extract information from the signals. In the simple learning process we find conditions for the risk-dominant and the non-risk-dominant equilibria to emerge alone in the long run. Furthermore, we find that the two equilibria can coexist in the long run. In contrast, the coexistence of the two equilibria is the only limit distribution under the maximum likelihood estimation learning algorithm. We also analyze the long-run equilibria of other 2x2 symmetric games under imperfect monitoring.

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r)/r times the stake with probability w and losing the stake with probability 1 − w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly with suitably chosen stakes. This problem has recently been extended in a unifying framework to account for limited playing time as well as future discounting, under which bold play is known to be optimal provided that w ≤ ½ ≤ r. This paper is concerned with a further extension of the Dubins-Savage gambling problem involving time-dependent parameters, and shows that bold play not only maximizes the probability of reaching the goal, but also stochastically minimizes the number of plays needed to reach the goal. As a result, bold play also maximizes the expected utility, where the utility at the goal is only required to be monotone decreasing with respect to the number of plays needed to reach the goal. It is further noted that bold play remains optimal even when the time-dependent parameters are random.

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

The Vardi casino with parameter 0 < c < 1 consists of infinitely many tables indexed by their odds, each of which returns the same (negative) expected winnings -c per dollar. A gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes and tables (odds). The optimal strategy is derived explicitly subject to the constraint that the gambler is allowed to play only a given finite number of times. Some properties of the optimal strategy are also discussed.

This paper is devoted to the study of two-person zero-sum games for continuous-time jump Markov processes with a discounted payoff criterion. The state and action spaces are all Polish spaces, the transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. We give conditions on the game's primitive data under which the existence of a solution to the Shapley equation is ensured. Then, from the Shapley equation, we obtain the existence of the value of the game and of a pair of optimal stationary strategies using the extended infinitesimal operator associated with the transition function of a possibly nonhomogeneous continuous-time jump Markov process. We also provide a recursive way of computing (or at least approximating) the value of the game. Moreover, we present a ‘martingale characterization’ of a pair of optimal stationary strategies. Finally, we apply our results to a controlled birth and death system and a Schlögl first model, and then we use controlled Potlach processes to illustrate our conditions.

In this paper we investigate a class of N-person nonconstant sum red-and-black games with bet-dependent win probability functions. We assume that N players and a gambling house are engaged in a game played in stages, where the player's probability of winning at each stage is a function f of the ratio of his bet to the sum of all the players' bets. However, at each stage of the game there is a positive probability that all the players lose and the gambling house wins their bets. We prove that if the win probability function is super-additive and it satisfies f(s)f(t)≤f(st), then a bold strategy is optimal for all players.

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r)/r times the stake with probability w and losing the stake with probability 1 - w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes. This problem has been extended in several directions to account for limited playing time or future discounting. We propose a unifying framework that covers these extensions, and prove that bold play is optimal provided that w ≤ ½ ≤ r. We also show that this condition is in fact necessary for bold play to be optimal subject to the constraint of limited playing time.

A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.

Christ and Avi-Itzhak (2002) analyzed a queueing system with two competing servers who determine their service rates so as to optimize their individual utilities. The system is formulated as a two-person game; Christ and Avi-Itzhak proved the existence of a unique Nash equilibrium which is symmetric. In this paper, we explore globally optimal solutions. We prove that the unique Nash equilibrium is generally strictly inferior to a globally optimal solution and that optimal solutions are symmetric and require the servers to adopt service rates that are smaller than those occurring in equilibrium. Furthermore, given a symmetric globally optimal solution, we show how to impose linear penalties on the service rates so that the given optimal solution becomes a unique Nash equilibrium. When service rates are not observable, we show how the same effect is achieved by imposing linear penalties on a corresponding signal.

In this paper a two-person red-and-black game is investigated. We suppose that, at every stage of the game, player I's win probability, f, is a function of the ratio of his bet to the sum of both players' bets. Two results are given: (i) if f is convex then a bold strategy is optimal for player I when player II plays timidly; and (ii) if f satisfies f(s)f(t) ≤ f(st) then a timid strategy is optimal for player II when player I plays boldly. These two results extend two formulations of red-and-black games proposed by Pontiggia (2005), and also provide a sufficient condition to ensure that the profile (bold, timid) is the unique Nash equilibrium for players I and II. Finally, we give a counterexample to Pontiggia's conjecture about a proportional N-person red-and-black game.

To each filter ℱ on ω, a certain linear subalgebra A(ℱ) of Rω, the countable product of lines, is assigned. This algebra is shown to have many interesting topological properties, depending on the properties of the filter ℱ. For example, if ℱ is a free ultrafilter, then A(ℱ) is a Baire subalgebra of ℱω for which the game OF introduced by Tkachenko is undetermined (this resolves a problem of Hernández, Robbie and Tkachenko); and if ℱ1 and ℱ2 are two free filters on ω that are not near coherent (such filters exist under Martin's Axiom), then A (ℱ1) and A(ℱ2) are two o-bounded and OF-undetermined subalgebras of ℱω whose product A(ℱ1) × A(ℱ2) is OF-determined and not o-bounded (this resolves a problem of Tkachenko). It is also shown that the statement that the product of two o-bounded subrings of ℱω is o-bounded is equivalent to the set-theoretic principle NCF (Near Coherence of Filters); this suggests that Tkachenko's question on the productivity of the class of o-bounded topological groups may be undecidable in ZFC.

In this paper we study players' long-run behaviors in evolutionary coordination games with imperfect monitoring. In each time period, signals corresponding to players' underlying actions, instead of the actions themselves, are observed. A boundedly rational quasi-Bayesian learning process is proposed to extract information from the realized signals. We find that players' long-run behaviors depend not only on the correlations between actions and signals, but on the initial probabilities of risk-dominant and non-risk-dominant equilibria being chosen. The conditions under which risk-dominant equilibrium, non-risk-dominant equilibrium, and the coexistence of both equilibria emerges in the long run are shown. In some situations, the number of limiting distributions grows unboundedly as the population size grows to infinity.

In 2002, Benjamin and Goldman gave a complete solution to a variant of the two-player card game Le Her. We extend their result by giving optimal strategies for the authentic version played with a deck consisting of arbitrary numbers of suits and denominations. Additionally, we show that the player who has the advantage in the game when one standard deck is used does not have the advantage if nineteen or more standard decks are used.

This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a two-sided change in the constant drift by means of a stopping rule. As a performance measure, an extended Lorden criterion is used. The goal is to minimize the worst-case detection delay subject to a constraint in the frequency of false alarms. In a companion paper, attention is drawn to a first category of 2-CUSUM rules for which the harmonic mean rule holds. It is further seen that a special class of 2-CUSUM stopping rules within this category, called drift equalizer rules, perform strictly better than non-equalizer rules, according to this specific performance measure.

For two-person zero-sum games, where the probability of each player winning is a continuous function of time and is known to both players, the mutually optimal strategy for proposing and accepting a doubling of the game value is known. We present an algorithm for deriving the optimal doubling strategy of a player who is aware of the suboptimal strategy followed by the opponent. We also present numerical results about the magnitude of the benefits; the results support the claim that repeated application of the algorithm by both players leads to the mutually optimal strategy.

In this paper, we study two-person nonzero-sum games for continuous-time Markov chains with discounted payoff criteria and Borel action spaces. The transition rates are possibly unbounded, and the payoff functions might have neither upper nor lower bounds. We give conditions that ensure the existence of Nash equilibria in stationary strategies. For the zero-sum case, we prove the existence of the value of the game, and also provide a recursive way to compute it, or at least to approximate it. Our results are applied to a controlled queueing system. We also show that if the transition rates are uniformly bounded, then a continuous-time game is equivalent, in a suitable sense, to a discrete-time Markov game.

We consider a service system (Q S) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data.