Paper # Author Title
Received auction theory prescribes that a reserve price which maximizes expected profit should be no less than the seller's own value for the auctioned object. In contrast, a common empirical observation is that many auctions have reserve prices set below seller's values, even at zero. This paper revisits the theory to find a potential resolution of the puzzle for second-price auctions. The main result is that an optimal reserve price may be less than the seller's value if bidders are risk averse and have interdependent values. Moreover, the resulting outcome may be arbitrarily close to that of an auction that has no reserve price, an absolute auction. Download Paper
We establish conditions under which an English auction for an indivisible risky asset has an efficient ex post equilibrium when the bidders are heterogeneous in both their exposures to, and their attitudes toward, the ensuing risk the asset will generate for the winning bidder. Each bidder's privately known type is unidimensional, but may affect both his risk attitude and the expected value of the asset's return to the winner. An ex post equilibrium in which the winning bidder has the largest willingness to pay for the asset exists if two conditions hold: each bidder's marginal utility of income is log-supermodular, and the vector-valued function mapping the type vector into the bidders' expected values for the asset satisfies a weighted average crossing condition. However, this equilibrium need not be efficient. We show that it is efficient if each bidder's expected value for the asset is nonincreasing in the types of the other bidders, or if the bidders exhibit nonincreasing absolute risk aversion, or if the asset is riskless. Download Paper
This paper concerns multistage games, with and without discounting, in which each player can increase the level of an action over time so as to increase the other players’ future payoffs. An action profile is achievable if it is the limit point of a subgame perfect equilibrium path. Necessary conditions are derived for achievability under relatively general conditions. They imply that any efficient profile that is approximately achievable must be in the core of the underlying coalitional game. In some but not all games with discounting, the necessary conditions for achievability are also sufficient for a profile to be the limit of achievable profiles as the period length shrinks to zero. Consequently, in these games when the period length is very short, (i) the set of achievable profiles does not depend on the move structure; (ii) an efficient profile can be approximately achieved if and only if it is in the core; and (iii) any achievable profile can be achieved almost instantly. Download Paper
This paper analyzes the effects of buyer and seller risk aversion in first and second-price auctions. The setting is the classic one of symmetric and independent private values, with ex ante homogeneous bidders. However, the seller is able to optimally set the reserve price. In both auctions the seller’s optimal reserve price is shown to decrease in his own risk aversion, and more so in the first-price auction. Thus, greater seller risk aversion increases the ex post efficiency of both auctions, and especially that of the first-price auction. The seller’s optimal reserve price in the first-price, but not in the second-price, auction decreases in the buyers’ risk aversion. Thus, greater buyer risk aversion also increases the ex post efficiency of the first but not the second-price auction. At the interim stage, the first-price auction is preferred by all buyer types in a lower interval, as well as by the seller.  Download Paper
This paper studies a class of dynamic voluntary contribution games in a setting with discounting and neoclassical payoffs (differentiable, strictly concave in the public good, and quasilinear in the private good). An achievable profile is the limit point of a subgame perfect equilibrium path -- the ultimate cumulative contribution vector of the players. A profile is shown to be achievable only if it is in the undercore of the underlying coalitional game, i.e., the profile cannot be blocked by a coalition using a component-wise smaller profile. Conversely, if free-riding incentives are strong enough that contributing zero is a dominant strategy in the stage games, then any undercore profile is the limit of achievable profiles as the period length shrinks. Thus, in this case when the period length is very short, (i) the set of achievable contributions does not depend on whether the players can move simultaneously or only in a round-robin fashion; (ii) an efficient profile can be approximately achieved if and only if it is in the core of the underlying coalitional game; and (iii) any achievable profile can be achieved almost instantly. Download Paper
A product exhibits personal fit uncertainty when its consumers have idiosyncratic and uncertain values for it. Often a consumer can learn her long-run value quickly by obtaining the good for a trial period. Money back guarantees of satisfaction are commonly used to lower the cost to consumers of learning their values this way. Increasingly, however, consumers can instead learn about their values before they purchase by, e.g., reading product reviews or consulting experts. We study the effect on a firm’s optimal price and refund of this competing source of information. An efficient outcome would be achieved by setting the refund for a return equal to its salvage value. But a monopoly will, for some parameters, induce consumers to stay uninformed by promising a refund that is greater than the salvage value. This generates an inefficiently large number of returns, which the firm finds worthwhile in order to eliminate the information rents that consumers would obtain by becoming informed. This finding is consistent with the observation that for many products, money back guarantees are generous, as they commonly refund the entire, or almost the entire, purchase price of a product. Download Paper
A monotone game is a multistage game in which no player can lower her action in any period below its previous level. A motivation for the monotone games of this paper is dynamic voluntary contribution to a public project. Each player's utility is a strictly concave function of the public good, and quasilinear in the private good. The main result is a description of the limit points of (subgame perfect) equilibrium paths as the period length shrinks. The limiting set of such profiles is equal to the undercore of the underlying static game - the set of profiles that cannot be blocked by a coalition using a smaller profile.  A corollary is that the limiting set of achievable profiles does not depend on whether the players can move simultaneously or only in a round-robin fashion. The familiar core is the efficient subset of the undercore; hence, some but not all profiles that are efficient and individually rational can be nearly achieved when the period length is small. As the period length shrinks, any core profile can be achieved in a “twinkling of the eye” - neither real-time gradualism nor inefficiency are necessary. Download Paper
A buyer can learn her value for a returnable experience good by trying it out, with the option of returning the good for whatever refund the seller offers. Sellers tend to offer a "no questions asked" refund for such returns, a money back guarantee. The refund is often too generous, generating inefficiently high levels of returns. We present two versions of a model of a returnable goods market. In the Information Acquisition Model, consumers are ex ante identical and uninformed of their private values for the good. The firm then offers a generous refund in order to induce the consumers to learn their values by purchasing and trying the good out, rather than by doing costly research prior to purchasing. In the Screening Model, some consumers have negligible costs of becoming informed about their values prior to purchasing, and always do so; other consumers have prohibitive costs of acquiring pre-purchase information and always stay uninformed. The firm's optimal screening menu may then contain only a single contract, one that specifies a generous refund, and hence a high purchase price, in order to weaken the incentive constraint of the informed consumers. Download Paper
We base a contracting theory for a start-up firm on an agency model with observable but nonverifiable effort, and renegotiable contracts. Two essential restrictions on simple contracts are imposed: the entrepreneur must be given limited liability,and the investor's earnings must not decrease in the realized profit of the firm. Allmessage game contracts with pure strategy equilibria (and no third parties) areconsidered. Within this class of contracts/equilibria, and regardless of who has the renegotiating bargaining power, debt and convertible debt maximize the entrepreneur's incentives to exert effort. These contracts are optimal if the entrepreneur has the bargaining power in renegotiation. If the investor has the bargaining power, the same is true unless debt induces excessive effort. In the latter case, a non-debt simple contract achieves efficiency - the non-contractibility of effort does not lower welfare. Thus, when the non-contractibility of effort matters, our results mirror typical capital structure dynamics: an early use of debt claims, followed by a switch to equity-like claims. Download Paper
We base a contracting theory for a start-up firm on an agency model with observable but nonverifiable effort, and renegotiable contracts. Two essential restrictions on simple contracts are imposed: the entrepreneur must be given limited liability, and the investor's earnings must not decrease in the realized profit of the firm. All message game contracts with pure strategy equilibria (and no third parties) are considered. Within this class of contracts/equilibria, and regardless of who has the renegotiating bargaining power, debt and convertible debt maximize the entrepreneur's incentives to exert effort. These contracts are optimal if the entrepreneur has the bargaining power in renegotiation. If the investor has the bargaining power, the same is true unless debt induces excessive effort. In the latter case, a non-debt simple contract achieves efficiency - the non-contractibility of effort does not lower welfare. Thus, when the non-contractibility of effort matters, our results mirror typical capital structure dynamics: an early use of debt claims, followed by a switch to equity-like claims. Download Paper
We present three examples of finitely repeated games with public monitoring that have sequential equilibria in private strategies, i.e., strategies that depend on own past actions as well as public signals. Such private sequential equilibria can have features quite unlike those of the more familiar perfect public equilibria: (i) making a public signal less informative can create Pareto superior equilibrium outcomes; (ii) the equilibrium final-period action profile need not be a stage game equilibrium; and (iii) even if the stage game has a unique correlated (and hence Nash) equilibrium, the first-period action profile need not be a stage game equilibrium. Download Paper