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Non-existence of a pure-strategy Nash equilibrium is a persistent problem in oligopoly models where sellers compete in prices and quantities. At the heart of this problem are unserved customers who are still willing to visit a less preferred supplier (i.e., so-called spillover demand). This note develops a consumer behavior model with spillover demand. Within this model there is a class of demand specifications for which a pure-strategy Nash equilibrium exists. This price-quantity equilibrium is shown to coincide with the Bertrand price equilibrium.

We model heterogeneous bidders’ bidding decisions and derive equilibrium solutions under both discriminatory and uniform price auctions. Under the assumptions that bidders’ cost of funds is uniformly distributed, and the winning probability is a linear decreasing function of the quoted price, we show that a revenue maximizing issuer should consider bidders’ cost of funds and winning probability when choosing the optimal auction mechanism.

We derive a parametric restriction necessary and sufficient for an equilibrium to exist in the model of Lenkey (2014). Given this restriction, the equilibrium is unique, correcting findings in the original article that multiple equilibria may exist with advance disclosure.

Nash solutions for two-player bargaining problems with non-convexity are shown to be dictatorial selections of Nash product maximizers in recent literature. In this paper we show that these solutions are implementable as unique subgame perfect equilibrium payoff allocations of a sequential game.

I introduce an undominated coalition-proof Nash equilibrium that is unique and is able to select undominated Nash equilibria in a quasi-supermodular game with a strict single-crossing property and monotonic externalities.

We develop a multi-player lottery Blotto game in which each contested item can be valuated differently by involved agents and in which each contested item can be differently valuated from other contested items. We prove that the Blotto game with a finite number of agents has a Nash equilibrium. We characterize all Nash equilibria for the case of two agents.

We study auctions for business licenses, where only the winners can participate in Cournot competition. This is an example of a multi-unit auction with endogenous and interdependent values. We compare a pay-as-bid auction and a uniform price auction. Assuming that only the winners’ identity and their payments are announced after the auction, the former auction is advantageous in terms of auction revenue, and the latter is advantageous in terms of consumer surplus.

When bidders’ valuations are derived from a downstream market in which they may compete, the allocation to the firms with the lowest costs can differ from the allocation that maximizes the ex post valuations of the bidders. I consider the problem of auctioning two goods to bidders whose valuations for a good flexibly depend on their and their rival’s costs as well as the identity of the rival. I show that revealing the identities of winners through a sequential auction procedure leads to allocations in which bidders tend to have higher ex post valuations but also higher costs when compared to a simultaneous auction.

This paper investigates a seller’s bundling decision when he/she sells multiple objects to two potential buyers through auctions in which he/she can optimally set reserve prices. We show how the optimal reserve prices in bundled/separate auctions and the seller’s bundling decisions are affected by the correlations of the valuations across buyers and objects. Interestingly, when each object’s valuations are sufficiently negatively correlated across buyers, the seller’s revenue is higher under unbundling than under bundling. The result is in contrast with that obtained by Palfrey (1983), who showed that, without considering optimal reserve prices, the seller’s revenue is always higher under bundling regardless of correlations across either buyers or objects when there are two buyers.

We formulate the notion of a socially-maximal Nash equilibrium distribution in a large distributionalized game with traits, and show that a socially-maximal Nash equilibrium distribution exists. We also show that no-where equivalence condition is not only sufficient but also necessary to the existence of a socially-maximal Nash equilibrium in a class of large individualized games with traits.

Most of the heterogeneous firm studies use the Pareto distribution of productivity. This paper identifies the Pareto parameter by relying on guidance from monopolistic competition models. The application to Korean export industries reports that the dispersion of productivity varies across industries.

When it is uncertain that a merger will have a cost-reducing efficiency and, should it materialize, the efficiency is private information to the merged firm, equilibrium prices may be at a level corresponding to the absence of the efficiency. In that case, none of the efficiency is passed to consumers.

The Diamond paradox (Diamond, 1971) asserts that in a market for a homogeneous good, if all consumers have positive search costs and search sequentially, then the unique equilibrium price is the monopoly price. I show that any finitely repeated version of this search game may support competitive prices.

We consider an infinite horizon game in which players have heterogeneous beliefs about future endowments process and trade with each other. Focusing on Pareto-efficient subgame-perfect equilibria, we find that unlike dynamic general equilibrium models, every player’s consumption is bounded away from zero. That is, every player survives in a Pareto-efficient subgame-perfect equilibrium.

Agents invest their available time in projects that generate profit. A mechanism divides the profit generated between these agents, depending on their allocation of time and the total profit of every project. We study mechanisms that incentivize agents to contribute their time to a level that results in the maximal aggregate profit at the strong Nash equilibrium (coalitional efficiency). Our finding is the characterization of all mechanisms that satisfy coalitional efficiency. These mechanisms depends on the type of available projects, their interconnectedness and expand earlier mechanisms that are non-bossy.

We consider second-price auctions with participation costs and investigate the revenue effects of a resale possibility. There are two potential bidders whose valuations are independently and identically distributed, and each bidder must incur a participation cost to bid in the auction. Due to these costs, there may be a resale opportunity in an otherwise efficient second-price auction. We show that when values are drawn from a uniform distribution, resale increases (decreases) entry of the lower-(higher-)cost bidder and decreases the original seller’s expected revenue.

We extend the multi-item auction model of Talman and Yang (2008) and Andersson et al. (2015) by considering both unit-demand bidders with budget constraints and price controls on bidding items. Due to these budget and price restrictions, a Walrasian equilibrium generally fails to exist. To achieve efficiency, we propose a rationed equilibrium whose allocation is in the core. We also construct an ascending auction to find the proposed rationed equilibrium in (pseudo-)polynomial time.

Fu et al. (2016) introduced a stability concept for labor markets with job security. We show that their proposed outcomes form Nash equilibria of an auction where firms compete for workers. This parallels literature on stable outcomes and similar auctions, and yields new price of anarchy bounds.

This paper studies the degrees of equilibrium competition in three common forms of auctions with costly participation, and shows that, when bidders' valuation distribution is concave, there is a simple condition to rank the equilibrium competition of those auctions. It also investigates how the results are related to stochastic ordering of bidders' valuation distributions, and provides some illustrative examples.

This paper shows that an important class of evolutionary game - theoretic models ? those in which agents form beliefs about the behavior of others on the basis of samples they observe from the population ? violate a property ? monotonicity ? that many authors have argued any well-specified evolutionary model should possess. A stripped down evolutionary model that closely follows Young?s (Econometrica, 1993) pioneering model is developed, and a theorem is presented showing that the sampling process on which beliefs are based necessarily generates a violation of monotonicity. (Loosely, a violation of monotonicity means that the population proportion of a "less fit" strategy grows more quickly than the population proportion of a "more fit" strategy.) A numerical example illustrating this violation is also presented. This violation does not imply that models with sampling-based beliefs are defective or implausible in any way. On the contrary, these models are built on carefully articulated and justified microfoundations. What this paper demonstrates is that analyses that begin with an assumption of monotonicity, without the specifics of the dynamic process explicitly stated, are not applicable to all evolutionary models.

This paper characterizes cartel pricing dynamics when consumers have reference-dependent preferences. Firms have a common discount factor unknown to consumers and a common cost i.i.d. over time. Consumers observe prices over time and update their expectations about firms’ ability to collude, which affects consumers’ price expectations. Reference-dependent preferences make consumers lose utility when the actual price is higher than the expected one, which forces colluding firms to raise prices alongside consumers’ price expectations. This increasing price path is capped by the price arising when consumers are sure that firms collude.

This paper is concerned with economic analysis of first-price sealed-bid auctions with risk averse bidders. The identification is based on exogenous variations in the number of bidders across auctions. We present a shape constrained estimator of the bidding function, which satisfies the theoretical properties of passing through origin, positivity and monotonicity. The underlying utility function and bidder value distribution are readily obtained from the estimated bidding function. Monte Carlo simulations demonstrate good performance of the proposed estimator.

We study all-pay auctions with winners’ reimbursement. This study is conducted under incomplete information with non-linear effort functions. We show that when the cost of effort is concave, the expected revenue in all-pay auctions with reimbursement is higher than the expected revenue in all-pay auctions without reimbursement. However, when the cost of effort is convex, the expected revenue in all-pay auctions without reimbursement is higher. The results of this paper supply a justification to apply reimbursement in contests as part of a mechanism to increase expected revenue. This is especially true in contests where applying new technologies results in decreasing the marginal costs of efforts.

In a large distributional game, we show that an action distribution is a Nash equilibrium action distribution if and only if for any subset of actions, the proportion of players playing actions within the subset is no more than the proportion of players having their best response in it. We also show that given an action set, a symmetric Nash equilibrium action distribution in every atomless large distributional game with the given action set can be characterized if and only if the action set is at most countable.

We consider sequential auction of two identical objects under the second-price rule when reserves are set optimally. We find that bidders bid more in the sequential auction when they are risk averse (CARA) than when they are not. The expected revenue is higher under the Vickrey rule than under the uniform-price rule. Hence the sequential auction generates higher expected revenue than both the Vickrey auction and uniform-price auction when bidders have constant absolute risk aversion.

A spiteful agent losing an auction experiences negative emotions towards the winner. In line with previous experimental findings I show that the symmetric equilibrium bid of a spiteful agent in third price auctions is higher than the risk neural Nash equilibrium bid. Furthermore, it can be shown that spiteful agents bid the same way agents would who anticipate to regret their decision in equilibrium.

We show that each mixed strategy of a normal-form game can be reformulated as a pure strategy of an induced Bayesian game. Moreover, a normal-form game has a Pareto undominated mixed-strategy Nash equilibrium if and only if its induced Bayesian game has a pure-strategy Bayesian Nash equilibrium. By relying on the existence result of Pareto undominated pure-strategy Nash equilibrium in Bayesian games in Fu and Yu (2015), we also show that every normal-form game has a Pareto undominated mixed-strategy Nash equilibrium.

Building on Ma (1994) and Fujinaka and Wakayama (2018) , this paper characterizes the top trading cycles rule in housing markets without the help of strategy-proofness, by the following two groups of axioms: individual rationality, Pareto efficiency, rank monotonicity; individual rationality, endowments-swapping-proofness, rank monotonicity.

A number of contestants randomly arrives according to a Poisson distribution. Under population uncertainty, a contest designer considers whether or not to reveal the number of players to bidders. We show that for the expected revenue, the two policies are equivalent, but for the expected winner’s payment, the revealing policy dominates if the expected number of bidders is sufficiently large, whereas the concealing policy dominates if it is low.

This work refines the notion of Nash equilibrium in the case of symmetric strictly competitive games. We define a (complete and typically intransitive) binary relation allowing to identify the so-called latent actions, for which there exists a maximal tree whose nodes are all preferred to the considered action. We prove the existence of patent Nash equilibria (obtained after iterated elimination of latent actions) and then describe the configurations that may arise when the two players have four (or less) actions available.

Network formation games (Myerson, 1991) typically present a multiplicity of Nash equilibria. Some of them are such that mutually beneficial links are not formed, thus inducing networks that are not pairwise stable. We offer an equilibrium refinement for this class of games which naturally involves pairwise stability while guaranteeing admissibility.

We investigate the effects of passive backward acquisitions in their efficient upstream supplier on downstream firms’ ability to collude in a dynamic game of price competition with homogeneous goods. We find that passive backward acquisitions impede downstream collusion. The main driver of our finding is that a passive backward acquisition secures an acquirer from zero continuation profits after a breakdown of collusion. This anti-collusive effect cannot be outweighed by a lower collusive price that is set by the cartel to increase the acquirer’s profit from its claim on the upstream margin.

We re-examine the Nash bargaining solution when an upstream and N downstream firms bargain over a linear input price with unobservable contracts. We show that the profit sharing rule is given by a simple and instructive formula which depends on the parties’ disagreement payoffs, the profit weights in the Nash-product and the elasticity of derived demand. A downstream firm’s profit share increases in the equilibrium derived demand elasticity which in turn depends on the final goods’ demand elasticity.

We investigate the effect of a ban on third-degree price discrimination on the sustainability of collusion. We build a model with two firms that may be able to discriminate between two consumer groups. Two cases are analyzed: (i) Best-response symmetries so that profits in the static Nash equilibrium are higher if price discrimination is allowed. (ii) Best-response asymmetries so that profits in the static Nash equilibrium are lower if price discrimination is allowed. In both cases, firms' discount factor has to be higher in order to sustain collusion in grim-trigger strategies under price discrimination than under uniform pricing.

We construct both monotonic and non-monotonic symmetric Bayesian Nash equilibria for a two-player all-pay contest with binary types and correlated information structures. We also employ a class of parametric distributions to illustrate our equilibrium construction explicitly and to derive some comparative statics results.

“Overbidding” with respect to risk-neutral Nash predictions in first-price auction experiments has been consistently reported in the literature. One possible explanation for overbidding is that participants in these experiments may try to avoid regret induced by the knowledge of winning bids in case they do not win these auctions. Such considerations may drive bidders to bid aggressively in first-price auctions. We test whether differences in how auction outcomes are revealed produces systematic differences in bidding. In our design, where individuals bid against pre-programmed computers, differences in revelation of winning bids, does not produce significant treatment differences. Our results are in contrast to previous experiments, which report systematic treatment differences based on whether winning bids are revealed or not.

We consider a symmetric two-player common-value setting where each player gets a private signal about the object value. We show that for some parameter values the equilibrium revenue can be higher in a Tullock contest than in the standard auctions.

We study the benchmark independent private value auction setting when bidders have endogenously determined budgets. Before bidding, a bidder decides how much money she will borrow. Bidders incur a cost to borrowing. We show that bidders are indifferent between participating in a first-price, second-price and all-pay auction. The all-pay auction gives higher revenue than the first-price auction, which gives higher revenue than the second price auction. In addition, when the distribution of values satisfies the monotone hazard rate condition, the revenue maximizing auction is implemented by an all-pay auction with a suitably chosen reserve price.

We prove that a variant of the second price auction for the sale of a single good through a Bayesian incentive compatible mechanism that maximizes expected revenue of the seller is optimal when the type space is discrete. Moreover, we show that this variant is related to the widely used generalized second price auction mechanism in keyword-auctions for advertising, thus providing a theoretical justification for a practical tool.

When firms are heterogeneous, there may not exist a common collusive price that raises all firms’ profits. However it is shown that there always exists a common minimum price that raises all firms’ profits.

In the independent private values auction model I study the continuity properties of the optimal auction revenue as a function of the valuations or types distribution. I show that the optimal revenue is lowersemicontinuous for convergence in distribution. If the limit distribution is continuous, the optimal auction revenue is continuous in distribution as well.

A recent literature has found a positive relationship between the disproportionality of the electoral system and the convergence of parties’ positions. Such a relationship depends crucially on the assumption that voting is sincere. We show that, when voters are players in the game and not simply automata that vote for their favorite party, two policy-motivated parties always take extreme positions in equilibrium.

International audience; Even failed takeovers can identify undervalued target firms. We find that compared to financial bidders, strategic buyers have a greater lasting valuation effect on the targets. Strategic bidders thus appear to be superior in identifying undervalued targets. JEL Classification: G30, G32, G34

It is well-known that in the wallet game with two bidders, bidding twice the (individual) signal is an equilibrium. We prove that this strategy is never an equilibrium in a Japanese–English auction once discrete bid levels are introduced; we also discuss the implications of this result.

This paper extends the Fishman (1988) model of preemptive jump bidding in private value takeover auction to auctions with common and affiliated values. We show that in the presence of a common value component, jump bidding equilibrium results in higher social surplus. Furthermore, we show that, under certain conditions, allowing for preemptive jump bidding in common value auctions results in ex-ante Pareto improvement.

This paper characterizes equilibrium bidding behavior in a multi-unit uniform-price auction. As posited by Noussair (1995) and Engelbrecht-Wiggans and Kahn (1998), when bidders demand two units and valuations are independent, equilibrium bidding behaviour entails bidding their values for the first unit, and bidding below their valuations for the second unit. We identify some errors in the analysis of Noussair (1995) and Engelbrecht-Wiggans and Kahn (1998), and provide a characterization of equilibrium behavior which involves a threshold value that separates bidders who bid zero for the second unit from those who shade their bids for the second unit.

We consider private-value auctions in the presence of exogenous participation costs and secret reserve prices endogenously set by sellers whose valuation of the item equals zero. We show that in first-price auctions there is an equilibrium for any arbitrary reserve price level. In second-price auctions, the only equilibrium with truthful bidding is a situation where no buyer enters and the seller chooses a reserve price that deters entry.

This paper deals with consumer markets where sales depend on production. Assuming that a firm’s sale is proportional to its production, it shows that a price leader will set a price which maximizes its industry’s revenues in markets with many small firms, independent of their cost functions.

We explore the link between informativeness of signals, stochastic dominance and equilibrium bids in a multi-unit auction with risk averse bidders. We show that for a particular class of signal distributions, there is a one-to-one relation between informativeness and conditional first-order stochastic dominance, so that a higher degree of informativeness in the signal-fundamental distribution corresponds to higher bids and therefore higher revenues. Finally, we consider a Walrasian bookbuilding auction and show that in such an auction, bidders may collude to push the price down to the seller's reservation price, regardless of the signals.

Restricting attention to players who use pure strategies, Tauman (2002) proves that in a k -price auction ( k ≥ 3 ) for every Nash equilibrium in which no player uses a weakly dominated strategy: (i) the bidder with the highest value wins the auction and (ii) pays a price higher than the second-highest value among the players, thereby generating more revenue for the seller than would occur in a first- or second-price auction. We show that these results do not necessarily hold when mixed strategies are allowed. In particular, we construct an equilibrium for k ≥ 4 in which the second-highest valued player wins the auction and makes an expected payment strictly less than her value. This equilibrium–which exists for any generic draw of player valuations–involves only one player using a nondegenerate mixed strategy, for which the amount of mixing can be made arbitrarily small.