The non-constant-sum Colonel Blotto game

The Colonel Blotto game is a two-player constant-sum game in which each player simultaneously distributes his fixed level of resources across a set of contests. In the traditional formulation of the Colonel Blotto game, the players?? resources are ??use it or lose it?? in the sense that any resources that are not allocated to one of the contests are forfeited. This article examines a non-constant-sum version of the Colonel Blotto game that relaxes this use it or lose it feature. We find that if the level of asymmetry between the players?? budgets is below a threshold, then there exists a one-to-one mapping from the unique set of equilibrium univariate marginal distribution functions in the constant-sum game to those in the non-constant-sum game. Once the asymmetry of the players?? budgets exceeds the threshold, this relationship breaks down and we construct a new equilibrium.     

Difference-form contest success functions and effort levels in contests

We examine players' equilibrium effort levels in a contest with difference-form contest success functions in which two players compete with each other to win a prize. We show the following. At the pure-strategy Nash equilibrium of a simultaneous-move game, and in the subgame-perfect equilibrium of a sequential-move game, only one of the players expends effort or neither player expends effort. If one player's composite strength is far greater than the other player's, only the player with greater composite strength expends effort whether they move simultaneously or sequentially. If the players' valuations for the prize and their marginal probabilities of winning at (0, 0) are sufficiently small, neither player expends effort whether they move simultaneously or sequentially.     

Contests with Alternative Public‐Good Prizes

I study contests in which a society of players compete, by expending irreversible effort, over which one of alternative prizes should be awarded to them by the decision maker. The prizes are public goods and/or public bads for the players. The players choose their effort levels simultaneously and independently. I define each player's valuation spread as the difference between his valuations for the two public‐good/public‐bad prizes. I establish that the players' equilibrium effort levels depend solely on their valuation spreads, and that the players never expend positive effort for both prizes in equilibrium. Further, I establish that in equilibrium only players with the widest positive valuation spread and players with the widest negative valuation spread expend positive effort. Finally, I establish that the equilibrium effort level expended for each prize and the equilibrium total effort level are determined only by the widest positive valuation spread and the widest negative valuation spread.     

The lifeboat problem

We study an all-pay contest with multiple identical prizes (“lifeboat seats”). Prizes are partitioned into subsets of prizes (“lifeboats”). Players play a two-stage game. First, each player chooses an element of the partition (“a lifeboat”). Then each player competes for a prize in the subset chosen (“a seat”). We characterize and compare the subgame perfect equilibria in which all players employ pure strategies or all players play identical mixed strategies in the first stage. The partitioning of prizes can lead to coordination failure when players employ nondegenerate mixed strategies. In these equilibria some rents are sheltered and rent dissipation is reduced.     

Strategic information exchange

We analyze a toy class of two-player repeated games with two-sided incomplete information. In our model, two players are facing independent decision problems and each of them holds information that is potentially valuable to the other player. We study to what extent, and how, information can be exchanged at equilibrium. We show that, provided oneʼs initial information is valuable to the other player, equilibria exist at which an arbitrary amount of information is exchanged at an arbitrary high rate. The construction relies on an indefinite, reciprocated, exchange.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

Communication in Repeated Games with Private Monitoring

The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at leasttwoother players.Journal of Economic LiteratureClassification Number: C73.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Rent-seeking with uncertain discriminatory power

We consider a two-player Tullock rent-seeking contest with uncertain discriminatory power in the contest success function. We examine the cases where both players are either informed or uninformed about the size of the discriminatory power, as well as the case where only one player has private information about it. We show that in all three cases the contest has a unique (Bayesian) Nash equilibrium. In each case we characterize key properties of the equilibrium.     

Alternating-offer games with final-offer arbitration

I analyze an alternating-offer model that integrates the common practice of having an arbitrator determine the outcomes if both playersʼ offers are rejected. I assume that the arbitrator uses final-offer arbitration (as in professional baseball). I find that if the arbitrator does not excessively favor one player, then the unique subgame-perfect equilibrium always coincides with the subgame-perfect equilibrium outcome in Rubinsteinʼs infinite-horizon alternating-offer game. However, if the arbitrator sufficiently favors the player making the initial offer, then delay occurs in equilibrium.     

Strategic information transmission networks

We study a model of multi-player communication. Privately informed decision makers have different preferences about the actions they take, and communicate to influence each others? actions in their favor. We prove that the equilibrium capability of any player to send a truthful message to a set of players depends not only on the preference composition of those players, but also on the number of players truthfully communicating with each one of them. We establish that the equilibrium welfare depends not only on the number of truthful messages sent in equilibrium, but also on how evenly truthful messages are distributed across decision makers.     

Arms Races and Negotiations

Two players simultaneously decide whether or not to acquire new weapons in an arms race game. Each player's type determines his propensity to arm. Types are private information, and are independently drawn from a continuous distribution. With probability close to one, the best outcome for each player is for neither to acquire new weapons (although each prefers to acquire new weapons if he thinks the opponent will). There is a small probability that a player is a dominant strategy type who always prefers to acquire new weapons. We find conditions under which the unique Bayesian–Nash equilibrium involves an arms race with probability one. However, if the probability that a player is a dominant strategy type is sufficiently small, then there is an equilibrium of the cheap-talk extension of the game where the probability of an arms race is close to zero.     

Experimenting and learning with localized direct communication

We report an experiment in which subjects may learn from each other. Specifically, a “queue” of players who are identically informed ex ante make decisions in sequence over two lotteries. Every player except the first in the queue observes (only) his immediate predecessor’s choice and payoff before making his own decision. In equilibrium decisions are identical from the first or second player onwards in all experimental conditions. However, complete adherence to equilibrium play is seldom observed in our experiment. We further analyze our data using a quantal response equilibrium approach and test for behavioral regularities related to base rate fallacy/conservatism bias, social conformity/rebelliousness, and preference for experimentation (preferring the lottery with potentially more information spillover value). Our estimations reveal a consistent preference for experimentation across conditions, and further analysis offers some support to our surmise that this behavioral regularity is due, in part, to an attempt to influence others behind in the queue.     

Equivalence and axiomatization of solutions for cooperative games with circular communication structure

We study cooperative games with transferable utility and limited cooperation possibilities. The focus is on communication structures where the set of players forms a circle, so that the possibilities of cooperation are represented by the connected sets of nodes of an undirected circular graph. Single-valued solutions are considered which are the average of specific marginal vectors. A marginal vector is deduced from a permutation on the player set and assigns as payoff to a player his marginal contribution when he joins his predecessors in the permutation. We compare the collection of all marginal vectors that are deduced from the permutations in which every player is connected to his immediate predecessor with the one deduced from the permutations in which every player is connected to at least one of his predecessors. The average of the first collection yields the average tree solution and the average of the second one is the Shapley value for augmenting systems. Although the two collections of marginal vectors are different and the second collection contains the first one, it turns out that both solutions coincide on the class of circular graph games. Further, an axiomatization of the solution is given using efficiency, linearity, some restricted dummy property, and some kind of symmetry.     

Gambling in contests

This paper presents a strategic model of risk-taking behavior in contests. Formally, we analyze an n-player winner-take-all contest in which each player decides when to stop a privately observed Brownian motion with drift. A player whose process reaches zero has to stop. The player with the highest stopping point wins. Unlike the explicit cost for a higher stopping time in a war of attrition, here, higher stopping times are riskier, because players can go bankrupt. We derive a closed-form solution of a Nash equilibrium outcome. In equilibrium, highest expected losses occur at an intermediate negative value of the drift.     

Information acquisition in conflicts

This paper considers incentives for information acquisition ahead of conflicts. First, we characterize the (unique) equilibrium of the all-pay auction between two players with one-sided asymmetric information where one player has private information about his valuation. Then, we use our results to study information acquisition prior to an all-pay auction. If the decision to acquire information is observable, but not the information received, one-sided asymmetric information can occur endogenously in equilibrium. Moreover, the cut-off values of the cost of information that determine equilibrium information acquisition are higher than those in the first best. Thus, information acquisition is excessive. In contrast, with open or covert information acquisition, the equilibrium cut-off values are as in the first best.     

Valuations and dynamics of negotiations

This paper analyzes three‐party negotiations in the presence of externalities. We obtain a closed‐form solution for the Markov perfect equilibrium of a multilateral non‐cooperative bargaining model, yielding an equilibrium value and dynamics of negotiations that are supported by experimental studies. Players’ values are monotonically increasing (or decreasing) in the amount of negative (or positive) externalities that they impose on others. Moreover, players’ values are continuous and piecewise linear on the worth of bilateral coalitions, and are inextricably related to their negotiation strategies: the equilibrium value is the Nash bargaining solution when no bilateral coalitions form; the Shapley value when all bilateral coalitions form; or the nucleolus, when either one bilateral coalition among “natural partners” or two bilateral coalitions including a “pivotal player” form.     

Contests with rank-order spillovers

This paper presents a unified framework for characterizing symmetric equilibrium in simultaneous move, two-player, rank-order contests with complete information, in which each player??s strategy generates direct or indirect affine ??spillover?? effects that depend on the rank-order of her decision variable. These effects arise in natural interpretations of a number of important economic environments, as well as in classic contests adapted to recent experimental and behavioral models where individuals exhibit inequality aversion or regret. We provide the closed-form solution for the symmetric Nash equilibria of this class of games, and show how it can be used to directly solve for equilibrium behavior in auctions, pricing games, tournaments, R&D races, models of litigation, and a host of other contests.