Negotiation games with unobservable mixed disagreement actions

A negotiation model combines an alternating offers bargaining game with a normal form stage game that determines players’ (interim) disagreement payoffs. Busch and Wen [Econometrica 63 (1995) 545] investigated this negotiation model under the assumption that players’ past mixed disagreement actions are observable. The question arises if this assumption is a substantial restriction. In this paper, we adopt the more plausible assumption that only the realizations of past mixed actions are observable, and find that the set of equilibrium payoffs shrinks, compared to that when mixed disagreement actions are observable. We precisely identify the effects of the unobservability and characterize the limiting set of equilibrium payoffs as the two players become sufficiently patient.     

Nash equilibrium reconsidered and an option for maximin

In this paper we will discuss some peculiarities of Nash equilibrium which are at odds with its standard applications: (a) the underlying dynamic interpretation, (b) the incentive independency if equilibria are mixed, and (c) the unprofitability if equilibrium and maximin are mixed and the game is 2-by-2. Maximin is proposed as an alternative solution concept in relevant situations.Institute of Economics, University of Hamburg, Von-Melle-Park 5, D-2000 Hamburg 13, Germany. This is a substantially revised version of the paper Anti-Nash and an Option for Maximin (Institute of Economics, University of Aarhus, Memo 1989-16), which the author presented at the Fourth Annual Congress of the European Economic Association, Augsburg, September 2–4, 1989.     

Cournot outcomes under Bertrand–Edgeworth competition with demand uncertainty

We provide new results for two-stage games in which firms make capacity investments when demand is uncertain, then, when demand is realized, compete in prices. We consider games with demand rationing schemes ranging from efficient to proportional rationing. In all cases, there is a subgame perfect equilibrium outcome coinciding with the outcome of the Cournot game with demand uncertainty if and only if (i) the fluctuation in absolute market size is small relative to the cost of capacity, or (ii) uncertainty is such that with high probability the market demand is very large and with the remaining probability the market demand is extremely small. Otherwise, equilibria involve mixed strategies. Further, we show under efficient rationing that condition (i) is sufficient for the unique equilibrium outcome to be an equilibrium outcome of the Cournot game with demand uncertainty.     

Multiplicity of mixed equilibria in mechanisms: A unified approach to exact and approximate implementation

We characterize full implementation of social choice sets in mixed-strategy Bayesian equilibrium. Our results concern both exact and virtual mixed implementation. For exact implementation, we identify a strengthening of Bayesian monotonicity, which we refer to as mixed Bayesian monotonicity. It is shown that, in economic environments with at least three agents, mixed Bayesian implementation is equivalent to mixed Bayesian monotonicity, incentive compatibility and closure. For implementing a social choice function, the case of two-agents is also covered by these conditions and mixed Bayesian monotonicity reduces to Bayesian monotonicity. Following parallel steps, mixed virtual implementation is shown to be equivalent to mixed virtual monotonicity, incentive compatibility and closure. The key condition, mixed virtual monotonicity, is argued to be very weak. In particular, it is weaker than Abreu–Matsushima’s measurability, thereby implying that: (1) virtual implementation in mixed Bayesian equilibrium is more permissive than virtual implementation in iteratively undominated strategies, and (2) non-regular mechanisms are essential for the implementation of rules in that gap.     

The Role of Upstream‐Downstream Competition on Bundling Decisions: Should Regulators Force Firms to Unbundle?

We develop an upstream–downstream model to analyze downstream firms' incentives to bundle. In our framework, the upstream firms are content providers (such as television stations) and the downstream firms are system operators (such as cable/satellite operators). We show that an a la carte regulation (i.e., a regulation that forces downstream firms to unbundle) leads to higher consumer surplus, if the unregulated equilibrium exhibits pure bundling (PB). Hence, our model predicts that in the television industry, which is mainly characterized by PB, an a la carte regulation will be beneficial for the consumers. If, on the other hand, the unregulated equilibrium is characterized by mixed bundling, then an a la carte regulation will increase consumer welfare provided that demand for multiple purchases is strong.     

Ambiguity in Electoral Competition

The paper proposes an explanation to why electoral competition induces parties to state ambiguous platforms even if voters dislike ambiguity. A platform is ambiguous if different voters may interpret it as different policy proposals. An ambiguous platform puts more or less emphasis on alternative policies so that it is more or less easily interpreted as one policy or the other. I suppose that a party can monitor exactly this platform design but cannot target its communications to individuals one by one. Each individual votes according to her understanding of the parties’ platforms but dislikes ambiguity. It is shown that this electoral competition has no Nash equilibrium. Nevertheless its max–min strategies are the optimal strategies of the Downsian game in mixed strategies. Furthermore, if parties behave prudently enough and if the voters aversion to ambiguity is small enough, these strategies do form an equilibrium.     

An extensive form solution to the adverse selection problem in principal/multi-agent environments

When agents have quasi-linear preferences, every incentive compatible social choice function can be implemented by a simple extensive form mechanism, even if agents are allowed to use mixed strategies. The second stage of the mechanism, which is used to elicit the agents' true preferences, is not reached in equilibrium; it gives agents strict dominant strategies, so equilibrium outcomes are not sensitive to agents' beliefs off the equilibrium path. This solves the multiple equilibrium problem of a principal facing several agents: the mechanism implements any solution to the principal's second best maximization problem. The specification of incentive compatibility constraints in the principal's problem presupposes a precise knowledge of the agents' beliefs. However, the above mechanism can be modified to implement the principal's second best (to within arbitrarily small perturbations of transfers), regardless of the agents' conditional beliefs. Received: 30 April 1997 / Accepted: 16 September 1997     

Mixed Duopoly with Subcontracting

Existing literature on mixed oligopoly focuses on competition among different types of firms but ignores their possible cooperation. We allow cooperation between a public firm and a private firm through subcontracting in a Hotelling mixed‐duopoly model. We find that when subcontracting is possible, the equilibrium without subcontracting is not socially optimal because subcontracting can lower total production costs. And if both firms engage in subcontracting, the existence of a public firm can guarantee the first best equilibrium, whether it is the low‐cost firm or not. But when a private firm is the low‐cost firm, it is more profitable for it to choose vertical foreclosure. And the consequent equilibrium is not socially desirable anymore. Copyright © 2014 John Wiley & Sons, Ltd.     

Stopping with congestion and private payoffs

This paper analyses a two-player stopping game with multiarmed bandits in which each player chooses between learning about the quality of her private risky arm and competing for the use of a single shared safe arm. The qualities of the players’ risky arms are independent. A player whose risky arm produces a success no longer competes for the safe arm. We assume that a player observes her opponent’s actions but not his realised payoffs. She is therefore never certain whether her opponent is still competing for the safe arm. When the players’ prior probabilities of success are sufficiently close, there exists no pure strategy equilibrium, and we characterise the unique mixed strategy equilibrium. Otherwise, the unique equilibrium is in pure strategies. The amount of experimentation performed in equilibrium is inefficiently low but, for many priors, higher than if successes are publicly observed.     

The relationship between mixed strategies and strategic groups

This paper identifies a theoretical basis for the existence of strategic groups. We represent the competitive marketplace by means of a game-theoretic model and motivate the existence of markets wherein the only Nash solution may be in mixed strategies. Given deterministic behavior on the part of the firms, we interpret this solution in terms of a strategic groups equilibrium. The major implication of this analysis is that strategic groups can exist even if there is no inherent asymmetry (in terms of skills) between firms.     

Invader strategies in the war of attrition with private information

Second price all-pay auctions (wars of attritions) have an evolutionarily stable equilibrium in pure strategies if valuations are private information. I show that for any level of uncertainty there exists a pure deviation strategy arbitrarily close to the equilibrium strategy such that for some valuations the equilibrium strategy has a selective disadvantage against the deviation if the population mainly plays the deviation strategy. I show that agents with those valuations would prefer to deviate even farther from the equilibrium strategy, if the population collectively uses the deviation strategy. I argue that in the Bayesian game studied here, a mass deviation can be caused by the entry of a small group of agents. The results provided in this paper imply that the equilibrium strategy is indeed unstable if one considers rare and independent mutations on the space of valuations. Numeric calculations indicate that the closer the deviation strategy to the equilibrium strategy, the more valuations are destabilizing.     

Gross substitutability of point-to-set correspondences

In this paper the notion of gross substitutability for the multi-valued case is studied. It is proved that, if in a pure exchange equilibrium model gross substitutability and some auxiliary conditions prevail, then (a) the set of equilibria is a Cartesian product of a convex set of equilibrium resource allocations and a convex cone of equilibrium prices; hence all equilibria are equiadvantageous for every trader; (b) the weak axiom of revealed preference holds in any equilibrium; (c) any equilibrium is stable with respect to reallocations of initial resources. Some situations in which Walras' law does not hold are considered as well.     

Trading equilibrium in a public good economy with smooth preferences and a mixed measure space of consumers

Lindahl and Nash equilibria are often used in the theory of public good. Shitovitz and Spiegel (1998) present an example of 2-person economy with one private good and one pure public good, where the core efficient Lindahl allocation does not Pareto dominate the (inefficient) Nash allocation. In this paper we introduce the new concept of Trading equilibrium for a general public good economy with smooth preferences and a mixed measure space of consumers. We obtain that this economy admits a unique Trading equilibrium. Moreover, the Trading equilibrium induces a core allocation that strictly Pareto dominates the Nash allocation.     

Coherent Dempster–Shafer equilibrium and ambiguous signals

This paper reappraises the Dempster–Shafer equilibrium, a novel solution concept for signaling games introduced by Eichberger and Kelsey (2004), and suggests a new refinement approach. It is demonstrated that if the types of the Sender–but not messages–are assumed to be ex-ante unambiguous, then the Receiver’s conditional Choquet preference derived by the Dempster–Shafer updating rule coincides with subjective expected utility. This property of the pessimistic updating rule narrows the pooling, but not separating, Dempster–Shafer equilibrium to be behaviorally equivalent to the perfect Bayesian equilibrium. Moreover, if one refines the separating Dempster–Shafer equilibrium à la Ryan (2002a) by imposing the belief persistence axiom, then no deviations from the perfect Bayesian equilibrium are feasible. To eliminate Ryan’s type of behavior, a less stringent refinement based on the notion of coherent beliefs is elaborated.     

The local stability of an open-loop Nash equilibrium in a finite horizon differential game

We propose a finite time differential game as a model for some economic processes and derive conditions for the Nash equilibrium solution to be locally asymptotically stable. We adopt the traditional ‘Cournot-reaction function’ notion of stability, which in our (continuous time) model becomes a function-to-function, or trajectory-to-trajectory, mapping. The conditions for stability seem to make economic sense. The equilibrium is less stable if the interaction terms in each period are large, if the game has a long duration, and if the discount rate is small.     

Mixed oligopoly, foreign firms, and location choice

We investigate a mixed market in which a state-owned, welfare-maximizing public firm competes against n domestic private firms and m foreign private firms which are all profit-maximizing. A circular city model with quantity-setting competition is employed. We find that the equilibrium location pattern depends on m. All private firms agglomerate in the unique equilibrium if m is zero or one. Two foreign firms induce differentiation between domestic and foreign private firms. More than two foreign firms yield differentiation among the foreign firms. Regardless of n and m, agglomeration of all domestic private firms appears in equilibrium. We provide several conditions in which eliminating the public firm from the market enhances social welfare. We extend the basic model and investigate three issues concerning multiple public firms, inefficiency of the public firm, and entries by private firms. We obtain some additional implications of welfare and equilibrium locations.     

Endogenous party structure

This paper proposes a model of two-party representative democracy on a single-dimensional political space, in which voters choose their parties in order to influence the parties’ choices of representative. After two candidates are selected as the median of each party’s support group, Nature determines the candidates’ relative likability (valence). Based on the candidates’ political positions and relative likability, voters vote for the preferable candidate without being tied to their party’s choice. We show that (1) there exists a nontrivial equilibrium under natural conditions, and (2) the equilibrium party border and the ex ante probabilities of the two-party candidates winning are sensitive to the distribution of voters. In particular, we show that if a party has a more concentrated subgroup, then the party tends to alienate its centrally located voters, and the party’s probability of winning the final election is reduced. Even if voter distribution is symmetric, an extremist party (from either side) can emerge as voters become more politically divided.     

Multiproduct price competition with heterogeneous consumers and nonconvex costs

This paper extends the oligopolistic model of price competition to environments with multiple goods, heterogeneous consumers, and arbitrary continuous cost functions. A Nash equilibrium in mixed strategies with an endogenous sharing rule is proven to exist. It is also shown that, in environments with fixed costs and constant marginal costs, all (symmetric and asymmetric) equilibria exhibit price dispersion across stores. Furthermore, the paper identifies scenarios in which prices will necessarily be random. In these markets, stores keep each other guessing because, given the fixed costs, they would incur a loss if their price strategies were anticipated and beaten by competitors. This is interpreted as an important economic feature that is possibly behind random price promotions such as weekly specials.     

On the preservation of deterministic cycles when some agents perceive them to be random fluctuations

Some recent equilibrium models give rise to complex but deterministic fluctuations. We modify the hypothesis of universal perfect foresight by injecting into the economy a nonnegligible fraction of less informed agents who optimize their expected utility with respect to the statistical distribution of prices in the deterministic dynamics. For the standard overlapping generations model with money (the 'Samuelson' case) it is proved that if the fraction of consumers with limited knowledge is sufficiently high, then all equilibrium cycles of period k ≥ 2 disappear. The global properties of the case of 2-cycles are studied in detail. A brief analysis of the 'classical' case is also given.     

Coupon Advertising Under Imperfect Price Information*

This paper studies sales promotions through coupons in an oligopoly under imperfect price information. Sellers can distribute either ordinary coupons, or coupon (price) advertising, or both types of coupons, at distant locations to attract consumers from their rivals' markets. A unique symmetric pure-strategy equilibrium exists where rebates and couponing intensity are always positive. In the ordinary-coupon equilibrium, prices, promotional efforts, and sellers' profits are higher than in the coupon-advertising equilibrium. However, if sellers are allowed to distribute both types of coupons, only coupon advertising is sent out in equilibrium.