Mixed strategy may outperform pure strategy: An initial study

In pure strategy meta-heuristics, only one search strategy is applied for all time. In mixed strategy meta-heuristics, each time one search strategy is chosen from a strategy pool with a probability and then is applied. An example is classical genetic algorithms, where either a mutation or crossover operator is chosen with a probability each time. The aim of this paper is to compare the performance between mixed strategy and pure strategy meta-heuristic algorithms. First an experimental study is implemented and results demonstrate that mixed strategy evolutionary algorithms may outperform pure strategy evolutionary algorithms on the 0-1 knapsack problem in up to 77.8% instances. Then Complementary Strategy Theorem is rigorously proven for applying mixed strategy at the population level. The theorem asserts that given two meta-heuristic algorithms where one uses pure strategy 1 and another uses pure strategy 2, the condition of pure strategy 2 being complementary to pure strategy 1 is sufficient and necessary if there exists a mixed strategy meta-heuristics derived from these two pure strategies and its expected number of generations to find an optimal solution is no more than that of using pure strategy 1 for any initial population, and less than that of using pure strategy 1 for some initial population

On the Existence of Equilibria in Discontinuous Games: three Counterexamples

We study whether we can weaken the conditions given in Reny [4] and still obtain existence of pure strategy Nash equilibria in quasiconcave normal form games, or, at least, existence of pure strategy ε−equilibria for all ε > 0. We show by examples that there are: 1. quasiconcave, payoff secure games without pure strategy ε−equilibria for small enough ε > 0 (and hence, without pure strategy Nash equilibria), 2. quasiconcave, reciprocally upper semicontinuous games without pure strategy ε−equilibria for small enough ε > 0, and 3. payoff secure games whose mixed extension is not payoff secure. The last example, due to Sion and Wolfe [6], also shows that nonquasiconcave games that are payoff secure and reciprocally upper semicontinuous may fail to have mixed strategy equilibria.N/

Correlation of Positive and Negative Reciprocity Fails to Confer an Evolutionary Advantage: Phase Transitions to Elementary Strategies

Economic experiments reveal that humans value cooperation and fairness. Punishing unfair behavior is therefore common, and according to the theory of strong reciprocity, it is also directly related to rewarding cooperative behavior. However, empirical data fail to confirm that positive and negative reciprocity are correlated. Inspired by this disagreement, we determine whether the combined application of reward and punishment is evolutionarily advantageous. We study a spatial public goods game, where in addition to the three elementary strategies of defection, rewarding, and punishment, a fourth strategy that combines the latter two competes for space. We find rich dynamical behavior that gives rise to intricate phase diagrams where continuous and discontinuous phase transitions occur in succession. Indirect territorial competition, spontaneous emergence of cyclic dominance, as well as divergent fluctuations of oscillations that terminate in an absorbing phase are observed. Yet, despite the high complexity of solutions, the combined strategy can survive only in very narrow and unrealistic parameter regions. Elementary strategies, either in pure or mixed phases, are much more common and likely to prevail. Our results highlight the importance of patterns and structure in human cooperation, which should be considered in future experiments

Dodging the steamroller: fundamentals versus the carry trade

Although, according to uncovered interest rate parity, exchange rates should move so as to prevent the carry trade being systematically profitable, there is a vast empirical literature demonstrating the opposite. High interest currencies more often tend to appreciate rather than depreciate, as noted by Fama (1983). In this paper, we treat volatility as the critical state variable and show that positive returns to the carry trade are overwhelmingly generated in the low-volatility "normal" state, whereas the high-volatility state is associated with lower returns or with losses as currencies revert to the long run level approximated by their mean real exchange rate - in other words, purchasing-power parity (PPP) tends to reassert itself, at least to some extent, during periods of turbulence. We confirm these results by comparing the returns from three possible monthly trading strategies: the carry trade, a strategy which is long the undervalued and short the overvalued currencies (the "fundamental" strategy) and a mixed strategy which involves switching from carry trade to fundamentals whenever volatility is in the top quartile. The mixed strategy generates positive returns greater than for either of the pure strategies

Solving Two-Person Zero-Sum Stochastic Games With Incomplete Information Using Learning Automata With Artificial Barriers

Learning automata (LA) with artificially absorbing barriers was a completely new horizon of research in the 1980s (Oommen, 1986). These new machines yielded properties that were previously unknown. More recently, absorbing barriers have been introduced in continuous estimator algorithms so that the proofs could follow a martingale property, as opposed to monotonicity (Zhang et al., 2014), (Zhang et al., 2015). However, the applications of LA with artificial barriers are almost nonexistent. In that regard, this article is pioneering in that it provides effective and accurate solutions to an extremely complex application domain, namely that of solving two-person zero-sum stochastic games that are provided with incomplete information. LA have been previously used (Sastry et al., 1994) to design algorithms capable of converging to the game's Nash equilibrium under limited information. Those algorithms have focused on the case where the saddle point of the game exists in a pure strategy. However, the majority of the LA algorithms used for games are absorbing in the probability simplex space, and thus, they converge to an exclusive choice of a single action. These LA are thus unable to converge to other mixed Nash equilibria when the game possesses no saddle point for a pure strategy. The pioneering contribution of this article is that we propose an LA solution that is able to converge to an optimal mixed Nash equilibrium even though there may be no saddle point when a pure strategy is invoked. The scheme, being of the linear reward-inaction ( $L_{R-I}$ ) paradigm, is in and of itself, absorbing. However, by incorporating artificial barriers, we prevent it from being ``stuck'' or getting absorbed in pure strategies. Unlike the linear reward-εpenalty ( $L_{R-ε P}$ ) scheme proposed by Lakshmivarahan and Narendra almost four decades ago, our new scheme achieves the same goal with much less parameter tuning and in a more elegant manner. This article includes the nontrial proofs of the theoretical results characterizing our scheme and also contains experimental verification that confirms our theoretical findings.acceptedVersio

Equilibrium strategies for overtaking free queueing networks under partial information

In this thesis, we are interested in finding equilibrium strategies for customers arriving at overtaking free queueing networks, only knowing partial information about the state of the system. The overtaking free condition does not allow customers to be overtaken by those behind them. We suppose that customers arrive at the system according to a Poisson process and that their service times at any queue are independent and exponentially distributed. Upon her arrival, the tagged customer is informed about the total number of customers in the system and chooses whether to join or not. Assuming that all customers follow the same strategy, the aim of the thesis is to find the equilibrium strategy that gives the maximum profit for any arriving customer. We show that such a strategy exists and is a pure or mixed threshold strategy. After analyzing the two-node tandem network and the multi-node tandem network, we focus on queueing networks with a branching structure, the so called tree networks. The work is extended with some numerical calculations, simulations and examples of non-overtaking free networks.In this thesis, we are interested in finding equilibrium strategies for customers arriving at overtaking free queueing networks, only knowing partial information about the state of the system. The overtaking free condition does not allow customers to be overtaken by those behind them. We suppose that customers arrive at the system according to a Poisson process and that their service times at any queue are independent and exponentially distributed. Upon her arrival, the tagged customer is informed about the total number of customers in the system and chooses whether to join or not. Assuming that all customers follow the same strategy, the aim of the thesis is to find the equilibrium strategy that gives the maximum profit for any arriving customer. We show that such a strategy exists and is a pure or mixed threshold strategy. After analyzing the two-node tandem network and the multi-node tandem network, we focus on queueing networks with a branching structure, the so called tree networks. The work is extended with some numerical calculations, simulations and examples of non-overtaking free networks

Partnership Dissolution and Proprietary Information

This paper studies different rules in dissolving a common value partnership where one partner holds proprietary information. In winner's bid auction (WBA) and loser's bid auction (LBA), there exists a unique mixed strategy equilibrium. ``Payoff equivalence'' is established in the sense that partners' expected payoffs are the same under the two auction formats. The informed partner benefits from an information rent while the uninformed suffers from an ``ownership's curse''. When cake-cutting mechanism (CCM) is applied, whether pure strategy equilibrium exists or not depends on the identity of the proposer. If the uninformed partner is the proposer, the informed partner receives an information rent in a pure strategy equilibrium. If the informed partner is the proposer, the asset value is shared equally between the two parties in a mixed strategy equilibrium. The paper then compares the information rents and prices offered for proprietary information by the three rules

List pricing and pure strategy outcomes in a Bertrand-Edgeworth duopoly

Non-existence of a pure strategy equilibrium in a Bertrand-Edgeworth duopoly model is analyzed. The standard model is modified to include a list pricing stage and a subsequent price discounting stage. Both firms first simultaneously choose a maximum list price and then decide to lower the price, or not, in a subsequent discounting stage. List pricing works as a credible commitment device that induces the pure strategy outcome. It is shown that for a general class of rationing rules there exists a sub-game perfect equilibrium that involves both firms playing pure strategies. This equilibrium payoff dominates any other sub-game perfect equilibrium of the game. Further unlike the dominant firm interpretation of a price leader, we show that the small firm may have incentives to commit to a low price and in this sense assume the role of a leader

Replication strategies and the evolution of cooperation by exploitation

Introducing the concept of replication strategies this paper studies the evolution of cooperation in populations of agents whose offspring follow a social strategy that is determined by a parent's replication strategy. Importantly, social and replication strategies may differ, thus allowing parents to construct their own social niche, defined by the behaviour of their offspring. We analyse the co-evolution of social and replication strategies in well-mixed and spatial populations. In well-mixed populations, cooperation-supporting equilibria can only exist if the transmission processes of social strategies and replication strategies are completely separate. In space, cooperation can evolve without complete separation of the timescales at which both strategy traits are propagated. Cooperation then evolves through the presence of offspring exploiting defectors whose presence and spatial arrangement can shield clusters of pure cooperators