Equilibrium in Hotelling's Model of Spatial Competition

This paper examines Hotelling's model of location with linear transportation cost. Existence of pure strategy subgame perfect equilibria in the infinitely repeated price game with fixed locations is proved. These subgame perfect equilibria have a stick and carrot structure. Given firm locations, there are discount factors sufficiently high that there is a subgame perfect equilibrium with a two-phase structure. Given the discount factors, there are stationary subgame perfect equilibria for a wide range of locations. However, for some pairs of location, no symmetric simple penal code exists, all subgame perfect profiles are nonstationary, and there is only one seller in the market in infinitely many periods.

We prove an equilibrium existence result for a class of games with an infinite number of strategies. Our theorem generalises an earlier result by Dasgupta and Maskin. We also identify conditions under which the limit of pure-strategy equilibria of a sequence of finite games is an equilibrium for the limit game. We apply this result to obtain new existence results for the multi-firm, i-dimensional version of Hotellings's location game. The techniques used suggest a technique for computing such equilibria. This paper studies games with discontinuous payoffs. Any game with a compact set of strategies can be approximated by a sequence of finite games. We study the correspondence mapping each approximation to the limit game to its mixed-strategy equilibria and identify conditions under which the limit of such equilibria is an equilibrium for the limit game. When payoffs are discontinuous, this property will not hold in general. An equilibrium for the limit game may not exist or, if it does exist, the equilibrium correspondence may not be upper hemi-continuous. The topic is of concern to economists for two, rather different, reasons. The existence question is clearly important in its own right. In many games that interest economists, payoff functions fail to be continuous or even semi-continuous. Standard existence theorems (Debreu (1952), Glicksberg (1950), (1952), etc.), therefore, are inapplicable. Two classical examples are Bertrand's (1883) model of price competition and Hotelling's (1929) model of spatial competition. Recent work on auction theory (see Milgrom-Weber (1982)) and dynamic oligopoly models (surveyed in Fudenberg-Tirole (1983)) has generated many more instances. The second reason for interest in the equilibrium correspondence is pragmatic. Finite games are generally much more tractable than infinite games. When the equilibrium correspondence is upper hemi-continuous, the easiest way to find an equilibrium for an infinite game may be to calculate the equilibria of a sequence of finite games and take limits.1 The seminal paper on games with discontinuous payoffs is Dasgupta-Maskin (DM) (1986a). This paper proves an existence theorem under assumptions that are strictly weaker than DM's. The key difference between our approach and DM's is as follows. DM identify conditions under which the limit of equilibria of any sequence of approximating finite games is an equilibrium for the limit game. These conditions are quite restrictive. They may be violated even though dominant strategy equilibria exist for the limit game. Our approach to the existence problem is different. We consider a given sequence of approximating games and ask when the limit of equilibria for this particular sequence will be an equilibrium for the limit game.2

A Noncooperative Analysis of Hotelling's Location Game

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.

Learning a mixed strategy equilibrium in the laboratory

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.

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.

We assume a multistage oligopoly wherein a given number of innovators compete by selling their substitutive technologies. Each innovator sequentially and independently chooses how many licenses to sell, and subsequently, all licensees compete à la Cournot in the product market. We show that, in equilibrium, the total number of licensees grows exponentially with the number of innovators. In addition, this sequential outcome is also obtained as a subgame perfect Nash equilibrium in pure strategies of a game with endogenous timing. Interestingly, by extending the duopoly model of Badia et al. (Math Soc Sci 108:8–13, 2020) to the case of more than two innovators and exploring pure strategy equilibria instead of mixed strategy equilibria, we derive drastically different policy implications, in terms of patent regulations. Our results suggest that more competition in the upstream market (e.g., by relaxing patent protection against the appearance of similar technologies) tends to increase downstream competition and welfare instead of discouraging or delaying technology adoption. In addition, our analysis is extended to explore the strategic role of public investment in basic R &D.

Two players, A and B, bargain to divide a perfectly divisible pie. In a bargaining model with constant discount factors, $$\delta _A$$ and $$\delta _B$$ , we extend Rubinstein (Econometrica 50:97–110, 1982)’s alternating offers procedure to more general deterministic procedures, so that any player in any period can be the proposer. We show that each bargaining game with a deterministic procedure has a unique subgame perfect equilibrium (SPE) payoff outcome, which is efficient. Conversely, each efficient division of the pie can be supported as an SPE outcome by some procedure if $$\delta _A+\delta _B\ge 1$$ , while almost no division can ever be supported in SPE if $$\delta _A+\delta _B < 1$$ .

Equilibrium in a discrete Downsian model given a non-minimal valence advantage and linear loss functions

Buying the Option to say 'No'

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.

Lead, Follow or Cooperate? Endogenous Timing &amp; Cooperation in Symmetric Duopoly Games

Non-Zero-Sum Stopping Games in Discrete Time

UNDOMINATED CANDIDATE STRATEGIES UNDER ALTERNATIVE VOTING RULES