Existence Of Pure-strategy Equilibrium Research Articles

Generalized medians and electoral competition with valence

I establish conditions for existence of pure strategy equilibria in K-candidate Downsian electoral competition (K ≥ 2) with valence when the voting rule is monotonic, generalizing existing results to non-proper rules and possibly continuous electorates. The conditions are sufficient when K ≥ 2 and (essentially) necessary in the K = 2 candidate case. They compare the size of one candidate's valence advantage to the radius of a generalized median pivotal ball (P-ball). I flesh out the difference of this generalized median with a recent alternative which, in turn, I characterize both on the basis of a weaker median property and using pivotal hyperplanes.

Hybrid energy sharing considering network cost for prosumers in integrated energy systems

The integrated energy system (IES) is developed to enhance the flexibility and efficiency of prosumers in the energy system. However, the current energy sharing method normally focuses on the energy cost and ignores the network cost, which constitutes a quarter of the prosumers’ electricity bills. It impacts energy sharing significantly. This paper proposes a hybrid energy sharing strategy for prosumers considering the electric and thermal network cost, to reduce the total cost. Firstly, the total cost of the prosumer is modelled based on joint energy-network cost. Secondly, the impact of uncertainty from renewables on the total cost is evaluated via the conditional value at risk. Thirdly, the non-cooperative game among prosumers is modelled and the existence of pure strategy equilibrium is proved. Then, the iterated best response algorithm based on differential evolution using limited information is proposed to protect personal privacy. Finally, the rationality and effectiveness of the model and the optimization method are verified by a coupled electricity-heat IES. The results show that the proposed model can reduce the energy consumption cost of prosumers and improve the penetration of renewable energy.

The All‐Pay Auction with Nonmonotonic Payoff

I model innovation contests as an all‐pay auction in which it is possible not to achieve successful innovation despite costly R&D investments, and as a result, there is no winner. In such a case, the winning payoff turns out to be nonmonotonic in own bid. I derive the sufficient conditions for the existence of pure strategy equilibria, and fully characterize the nondegenerate mixed strategy equilibrium. In the mixed strategy equilibrium, the support of the low‐value bidder is not continuous, and both the high‐value and the low‐value bidders place an atom in the (distinct) lower bound of their respective support. Under symmetric valuation, both bidders place an atom at zero. These results can explain why one does not observe very low quality innovation in real life, or why even symmetric firms may stay out of an innovation contest.

Two-Sided Learning and the Ratchet Principle

I study a class of continuous-time games of learning and imperfect monitoring. A long-run player and a market share a common prior about the initial value of a Gaussian hidden state, and learn about its subsequent values by observing a noisy public signal. The long-run player can nevertheless control the evolution of this signal, and thus affect the market’s belief. The public signal has an additive structure, and noise is Brownian. I derive conditions for an ordinary differential equation to characterize equilibrium behavior in which the long-run player’s actions depend on the history of the game only through the market’s correct belief. Using these conditions, I demonstrate the existence of pure-strategy equilibria in Markov strategies for settings in which the long-run player’s flow utility is nonlinear. The central finding is a learning-driven ratchet principle affecting incentives. I illustrate the economic implications of this principle in applications to monetary policy, earnings management, and career concerns.

Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

Bargaining through Approval

The paper considers two-person bargaining under Approval Voting. It first proves the existence of pure strategy equilibria. Then it shows that this bargaining method ensures that both players obtain at least their mean utility level in equilibrium. Finally it proves that, provided that the players are partially honest, the mechanism triggers sincerity and ensures that no alternative Pareto dominates the outcome of the game.

Existence of equilibria in constrained discontinuous games

We establish the existence of pure strategy equilibria in games with discontinuous payoffs where the set of feasible actions of each player varies, also in a discontinuous fashion, as a function of the actions of the other players. Such games are used in modeling abstract economies and other games where players share common constraints. Our approach circumvents the difficulties that arise from the presence of discontinuities by modifying the original problem and allowing the players to use strategies that possibly lie outside their feasible sets. We then show that each modified game has \(\varepsilon \)-equilibria points. Under certain conditions, and as the extent of modification becomes smaller and smaller and \(\varepsilon \) approaches zero, the \(\varepsilon \)-equilibria points of the modified games will converge to a strategy profile that is an equilibrium of the original game. Hence, we obtain a set of sufficient conditions for the existence of pure equilibria of the original game. We apply our results to a number of classic games that have discontinuous payoffs and discontinuous constraint correspondences.

On the diffuseness of incomplete information game

We introduce the “relative diffuseness” assumption to characterize the differences between payoff-relevant and strategy-relevant diffuseness of information. Based on this assumption, the existence of pure strategy equilibria in games with incomplete information and general action spaces can be obtained. Moreover, we introduce a new notion of “undistinguishable purification” which strengthens the standard purification concept, and its existence follows from the relative diffuseness assumption.

Existence of Equilibria in Constrained Discontinuous Games

We establish the existence of pure strategy equilibria in games with discontinuous payoffs where the set of feasible actions of each player varies, also in a discontinuous fashion, as a function of the actions of the other players. Such games are used in modeling abstract economies and other games where players share common constraints. Our approach circumvents the difficulties that arise from the presence of discontinuities by modifying the original problem and allowing the players to use strategies that possibly lie outside their feasible sets. We then show that each modified game has epsilon-equilibria points. Under certain conditions, and as the extent of modification becomes smaller and smaller and approaches zero, the epsilon-equilibria points of the modified games will converge to a strategy profile that is an equilibrium of the original game. Hence, we obtain a set of sufficient conditions for the existence of pure equilibria of the original game. We apply our results to a number of classic games that have discontinuous payoffs and discontinuous constraint correspondences.

On the existence of pure-strategy equilibria in games with private information: A complete characterization

This paper reports a definitive resolution to the question of the existence of a pure-strategy Bayesian–Nash equilibrium in games with a finite number of players, each with a compact metric action set and private information. The resolution hinges on saturated spaces. If the individual spaces of information are saturated, there exists a pure-strategy equilibrium in such a game; and if there exists a pure-strategy equilibrium for the class of games under consideration and with uncountable action sets, the spaces of private information must be saturated. As such, the paper offers a complete characterization of a longstanding question, and offers another game-theoretic characterization of the saturation property, one that complements a recent result of Keisler–Sun (2009) on large non-anonymous games with complete information.

Power sharing and electoral equilibrium

This paper considers a model of (consensual) democracy where political parties engage first in electoral competition, and they share policy-making power afterward according with the votes gathered in the election. The paper uncovers the difficulties to guarantee stability in this institutional setting; and it provides a condition of symmetry on parties’ political motivations that ensures the existence of pure strategy equilibrium under a broad family of power sharing rules, ranging from fully proportional to winner-take-all. The equilibrium analysis shows that power sharing and ideology exert a centrifugal force on policy platforms that increases party polarization, with the paradoxical result that consensual democracies can actually lead to more radical electoral campaigns than winner-take-all.

Large games with a finite number of classes and Nash's `improved' proof

Nash (1951) proved the existence of mixed strategy Nash equilibria for finite games using Brouwer’s, rather than Kakutani’s, fixed point theorem. This paper adapts Nash’s (1951) proof to large games with a finite number of classes to establish the existence of pure strategy equilibria. The new approach sheds some light on the old results.

On existence of pure strategy equilibrium with endogenous income

It is well known that in a voting-over-income-taxation game, there exists no pure strategy equilibrium when voters’ incomes are exogenous (i.e., when individual voters make only consumption choices). I prove that in an environment with endogenous income (i.e., where individual voters make labor-leisure choices, in addition to consumption choices) and candidates propose marginally progressive taxes, it is possible to get existence of a pure-strategy equilibrium tax schedule.

A Direct Proof of the Existence of Pure Strategy Equilibria in Large Generalized Games with Atomic Players

Consider a game with a continuum of players where only a finite number of them are atomic. Objective functions and admissible strategies may depend on the actions chosen by atomic players and on aggregate information about the actions chosen by non-atomic players. Only atomic players are required to have convex sets of admissible strategies and quasi-concave objective functions. In this context, we prove the existence of pure strategy Nash equilibria, a result that extends Rath (1992, Theorem 2) to generalized games and gives a direct proof of a special case of Balder (1999, Theorem 2.1). Our proof has the merit of being simple, based only on standard fixed point arguments and finite dimensional real analysis.

Why saturated probability spaces are necessary

An atomless probability space ( Ω , A , P ) is said to have the saturation property for a probability measure μ on a product of Polish spaces X × Y if for every random element f of X whose law is marg X ( μ ) , there is a random element g of Y such that the law of ( f , g ) is μ. ( Ω , A , P ) is said to be saturated if it has the saturation property for every such μ. We show each of a number of desirable properties holds for every saturated probability space and fails for every non-saturated probability space. These include distributional properties of correspondences, such as convexity, closedness, compactness and preservation of upper semi-continuity, and the existence of pure strategy equilibria in games with many players. We also show that any probability space which has the saturation property for just one “good enough” measure, or which satisfies just one “good enough” instance of the desirable properties, must already be saturated. Our underlying themes are: (1) There are many desirable properties that hold for all saturated probability spaces but fail everywhere else; (2) Any probability space that out-performs the Lebesgue unit interval in almost any way at all is already saturated.

The All-Pay Auction with Non-Monotonic Payoff

We analyse an endogenous prize all-pay auction under complete information where it is possible that none of the bidders wins and the winning payoff becomes non-monotonic in own bid. We derive the conditions for the existence of pure strategy equilibria and fully characterize the unique mixed strategy equilibrium when pure strategy equilibria do not exist. The highvaluation bidder places two distinct mass points in his equilibrium mixed strategy and the equilibrium support of the low-valuation bidder is not continuous. Under common valuation, in the equilibrium mixed strategy both bidders place mass points at the same point of the support. Possible applications are discussed.

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 [D. Schmeidler, Equilibrium points of nonatomic games, J. Stat. Phys. 4 (1973) 295–300], Rashid [S. Rashid, Equilibrium points of non-atomic games: Asymptotic results, Econ. Letters 12 (1983) 7–10], Mas-Colell [A. Mas-Colell, On a theorem by Schmeidler, J. Math. Econ. 13 (1984) 201–206], Khan and Sun [M. Khan, Y. Sun, Non-cooperative games on hyperfinite Loeb spaces, J. Math. Econ. 31 (1999) 455–492] and Podczeck [K. Podczeck, On purification of measure-valued maps, Econ. Theory 38 (2009) 399–418]. 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.

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.

Minimum wages and welfare in a Hotelling duopsony

Two firms choose locations (non-wage job characteristics) on the interval [0, 1] prior to announcing wages at which they employ workers who are uniformly distributed; the (constant) marginal revenue products of workers may differ. Subgame perfect equilibria of the two-stage location-wage game are studied under laissez-faire and under a minimum wage regime. Up to a restriction for the existence of pure strategy equilibria, the imposition of a minimum wage is always welfare-improving because of its effect on non-wage job characteristics.

Mixed-strategy equilibria and strong purification for games with private and public information

This paper shows the existence of mixed-strategy equilibria for games with private and public information under general conditions. Under the additional assumptions of finiteness of action spaces and diffuseness and conditional independence of private information, a strong purification result is obtained for the mixed strategies in such games. As a corollary, the existence of pure-strategy equilibria follows.