Payoff Equivalence Research Articles

Contests with discontinuous payoffs

We consider a contest model with non-monotonic payoff and cost functions as introduced by Siegel (2014b). In this paper we generalize this set-up to also allow for cases in which payoffs and costs of the players can have some kind of discontinuities. Relevant real-life examples include open markets where firm decisions lead to discontinuous costs and competitions where, for instance, an extra reward is allocated to the contestant when a particular performance threshold is reached. Players compete by investing some effort in order to win one of several identical prizes. We show that one can relax the assumptions in Siegel (2009, 2014b) so that the Payoff Equivalence Theorem of Siegel still holds with the aforementioned discontinuities. We present relevant examples which show that the relaxed assumptions are indispensable, and therefore one cannot expect to refine the assumptions of Siegel more than the ones presented in this paper.

Payoff equivalence of efficient mechanisms in large matching markets

We study Pareto efficient mechanisms in matching markets when the number of agents is large and individual preferences are randomly drawn from a class of distributions, allowing for both common and idiosyncratic shocks. We provide a broad set of circumstances under which, as the market grows large, all Pareto efficient mechanisms---including top trading cycles (with an arbitrary ownership structure), serial dictatorship (with an arbitrary serial order), and their randomized variants---produce a distribution of agent utilities that in the limit coincides with the utilitarian upper bound. This implies that Pareto efficient mechanisms are uniformly asymptotically payoff equivalent ``up to the renaming of agents.'' Hence, when the conditions of our model are met, policy makers need not discriminate among Pareto efficient mechanisms based on the aggregate payoff distribution of participants.

Stretch It but Don’t Break It: The Hidden Cost of Contract Framing

Recent research suggests that loss-framed contracts are an effective instrument for principals to maximize the effort of their agents. Framing effects arise from defining quantity and quality thresholds that vary the salience of losses and gains even while preserving the payoff equivalence of the underlying contract. While plausible interpretations of prospect theory’s loss-aversion insight suggest that a loss frame would lead to more effort, we show that contract thresholds also exert a norm-framing effect on performance that can trump the impact of loss aversion. Loss framing therefore carries a risk: poorly selected thresholds may reduce effort. Principals may prefer to avoid this risk by offering contracts that impose no threshold at all.

Bargaining in dynamic markets

We study non-stationary markets in which traders are randomly matched to bargain over the price of a heterogeneous good or the terms of a partnership. The economy consists of a continuum of players drawn from a finite set of types. Players exogenously enter the market over time and exit upon trading. At every date, matching probabilities for each pair of types are determined by the endogenous distribution of trader types in the market. The balance of bargaining power at any stage depends on variations in potential gains from trade, the inflows of new traders, the structure of agreements at future dates, and the induced frequency of trading opportunities. We establish that an equilibrium always exists. Moreover, all equilibria that lead to the same evolution of the economy are payoff equivalent. However, we show that multiple self-fulfilling expectations about the trajectory of the economy, generating distinct equilibrium dynamics and payoffs, may coexist.

Participation and Unbiased Pricing in CDS Settlement Mechanisms

The centralized market for the settlement of credit default swaps (CDS), which governs more than $20 trillion’s worth of outstanding CDS contracts, has been criticized for underpricing the defaulted bonds that underlie the contracts. I take a mechanism design approach to the market for the settlement of CDS contracts and characterize robust settlement mechanisms that deliver unbiased prices for the underlying assets. All robust settlement mechanisms are the payoff equivalent of a posted price mechanism. Because forced participation in the settlement mechanism is not possible, my approach requires the development of a new notion of the core of games of incomplete information. This new notion can be applied to mechanism design environments in which side trades are allowed or when joining the mechanism is a cooperative decision.

A Note on Payoff Equivalence of the Volunteer's Dilemma and the Stag Hunt Game and Inferiority of Intermediate Thresholds

The launch of a public project requires "enough" support from a group of 'n' players, i.e., a certain threshold has to be passed. The players may be differently important for passing the threshold; they may have different costs of support and different benefits if the project is launched. If players have only binary decision sets (participate or not, vote approvingly or not) this game is called a Binary Threshold Public Goods game (BTPG). We compare the expected equilibrium payoffs in BTPGs with the same costs and benefits but different thresholds. Applying two principles of equilibrium selection, the least and the most demanding threshold, namely "one supporting player is sufficient" (Volunteer's Dilemma) and "support by all players is necessary" (Stag Hunt game) are payoff equivalent for all players. Compared with the Stag Hunt game, all intermediate thresholds are connected with Pareto-inferior payoffs.

Payoff Equivalence of Efficient Mechanisms in Large Matching Markets

We study Pareto efficient mechanisms in matching markets when the number of agents is large and individual preferences are randomly drawn from a class of distributions, allowing for both common and idiosyncratic shocks. We show that, as the market grows large, all Pareto efficient mechanisms -- including top trading cycles, serial dictatorship, and their randomized variants -- are uniformly asymptotically payoff equivalent "up to the renaming of agents," yielding the utilitarian upper bound in the limit. This result implies that, when the conditions of our model are met, policy makers need not discriminate among Pareto efficient mechanisms based on the aggregate payoff distribution of participants.

Payoff Equivalence of Efficient Mechanisms in Large Matching Markets

We study Pareto efficient mechanisms in matching markets when the number of agents is large and individual preferences are randomly drawn from a class of distributions, allowing for both common and idiosyncratic shocks. We show that, as the market grows large, all Pareto efficient mechanisms -- including top trading cycles, serial dictatorship, and their randomized variants -- are uniformly asymptotically payoff equivalent “up to the renaming of agents,” yielding the utilitarian upper bound in the limit. This result implies that, when the conditions of our model are met, policy makers need not discriminate among Pareto efficient mechanisms based on the aggregate payoff distribution of participants.

Stretch It But Don't Break It: The Hidden Risk of Contract Framing.

Recent research suggests that loss framed contracts are an effective instrument for principals to maximize the effort of their agents. Framing effects arise from defining thresholds that vary the salience of losses and gains while preserving payoff equivalence of the underlying contract. While under Prospect Theory a loss frame should lead to more effort we show that contract thresholds also exert a suggestive effect on performance that can trump the impact of loss aversion. Loss framing therefore carries a risk. As agents focus their effort choice on the expressed thresholds, poorly selected thresholds reduce effort and the principal might prefer offering a contract that does not impose a threshold at all. On the other hand, imposing demanding thresholds may push effort beyond levels predicted by Prospect Theory.

Pricing in position auctions and online advertising

Can search engines increase revenues by changing their position auctions? In this paper, I analyze position auctions with general pricing rules to answer this question. In these auctions, there are several items that are commonly ranked by bidders with unit demand. I show that revenues remain the same for position auctions with regular pricing rules where the price for an item depends on the bids of agents who win lower-ranked items. In addition, all of the bidders also have the same ex post payo ffs. I also show that regularity is a necessary condition to get ex post payoff equivalence.

Ambiguous beliefs and mechanism design

This paper develops a payoff equivalence theorem for mechanisms with ambiguity averse participants with preferences of the Maxmin Expected Utility (MEU) form (Gilboa and Schmeidler, 1989). We use our payoff equivalence result to explicitly characterize the revenue maximizing private value auction mechanism for agents with arbitrary forms of ambiguous beliefs. We also show that the revenue ranking between first and second price auctions is sensitive to the form of ambiguity aversion. Our payoff equivalence techniques allow us to study the constrained efficient, budget balanced bilateral trade mechanism and show that increased ambiguity improves the efficiency of the mechanism. In addition, we characterize the revenue maximizing, efficient bilateral trade mechanism and show that heightened ambiguity lowers ex ante budget deficits.

Payoff Equivalence Theorem with Rich Type Spaces

This paper deals with the payoff equivalence theorem. Krishna and Maenner established equivalence for finite-dimensional type space. By introducing a 'known-own-payoff type space' a la Bergemann and Morris, we establish the sufficient conditions that guarantee payoff equivalence even in the world of 'rich type spaces.'

Payoff Equivalence Theorem with Rich Type Spaces

Payoff Equivalence Theorem with Rich Type Spaces

Payoff Equivalence Theorem with Rich Type Spaces

Payoff Equivalence Theorem with Rich Type Spaces

Does timing of decisions in a mixed duopoly matter?

We determine the endogenous order of moves in a mixed price-setting duopoly. In contrast to the existing literature on mixed oligopolies we establish the payoff equivalence of the games with an exogenously given order of moves if the most plausible equilibrium is realized in the market. Hence, in this case it does not matter whether one becomes a leader or a follower. We also establish that replacing a private firm by a public firm in the standard Bertrand–Edgeworth game with capacity constraints increases social welfare and that a pure-strategy equilibrium always exists.

Bargaining in Stationary Networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain. Players who reach agreement are replaced by new players at the same positions in the network. We show that all equilibria are payoff equivalent. The payoffs and the set of agreement links converge as players become patient. Several new concepts—mutually estranged sets, partners, and shortage ratios—provide insights into the relative strengths of the positions in the network. We develop a procedure to determine the limit equilibrium payoffs for all players. Characterizations of equitable and nondiscriminatory networks are also obtained. (JEL C78, D85)

Pricing in Position Auctions and Online Advertising

Can search engines increase revenues by changing their position auctions? In this paper, I analyze position auctions with general pricing rules to answer this question. In these auctions, there are several items that are commonly ranked by bidders with unit demand. I show that revenues remain the same for position auctions with regular pricing rules where the price for an item depends on the bids of agents who win lower-ranked items. In addition, all of the bidders also have the same ex post payo ffs. I also show that regularity is a necessary condition to get ex post payoff equivalence.

On the uniqueness of Groves mechanisms and the payoff equivalence principle

Consider a standard mechanism design setting with quasi-linear preferences and private valuations. From Holmström (1979), we know that if the valuations are smooth with respect to types then any efficient, dominant strategy mechanism is in the class of Groves mechanisms. Here I show that, given regular assumptions on the primitives of the design problem, a weaker condition that includes the case of non-smooth valuations is sufficient and necessary for the uniqueness of Groves mechanisms among all efficient, dominant strategy mechanisms. This condition, which imposes a restriction on the behavior of the one-sided directional derivatives of the valuation functions with respect to individual types, is also shown to be sufficient and necessary to obtain the Payoff Equivalence principle for dominant strategy mechanisms whose choice rules are affine maximizers.

Private versus complete information in auctions

We show that, in several (but not all) standard auctions, (i) private independent values and (ii) complete information are payoff equivalent. The first-price auction generates the same revenue under (i) and (ii); the all-pay auction has lower revenue under (ii).

Payoff equivalence in sealed bid auctions and the dual theory of choice under risk

This paper considers single item auctions in the private values framework, with buyers whose preferences satisfy the axioms of Yaari’s [Econometrica 55 (1987) 95] dual theory of choice under risk. It is shown that when their valuations are independently distributed, risk averse buyers are indifferent among all the auctions contained in a large family of mechanisms that includes the standard auctions.