Games with Discontinuous Payoffs

We study Hotelling's two-stage model of spatial competition, in which two firms first simultaneously choose locations in the unit interval, then simultaneously choose prices. Under Hotelling's assumptions (uniform distribution of consumers, travel cost proportional to distance, inelastic demand of one unit by each consumer) the price-setting subgames possess equilibria in pure strategies for only a limited set of location pairs. Because of this problem (pointed out independently by Vickrey (1964) and d'Aspremont et al. (1979)), Hotelling's claim that there is an equilibrium of the two-stage game in which the firms locate close to each other is incorrect. A result of Dasgupta and Maskin (1986) guarantees that each price-setting subgame has an equilibrium in mixed strategies. We first study these mixed strategy equilibria. We are unable to provide a complete characterization of them, although we show that for a subset of location pairs all equilibria are of a certain type. We reduce the problem of finding an equilibrium of this type to that of solving three or fewer highly nonlinear equations. At each of a large number of location pairs we have computed approximate solutions to the system of equations. Next, we use our analytical results and computations to study the equilibrium location choices of the firms. There is a unique (up to symmetry) subgame perfect equilibrium in which the location choices of the firms are pure; in it, the firms locate 0.27 from the ends of the market. At this equilibrium, the support of the subgame equilibrium price strategy is the union of two short intervals. Most of the probability weight is in the upper interval, so that this strategy is reminiscent of occasional sales by the firms. We also find a subgame perfect equilibrium in which each firm uses a mixed strategy in locations. In fact, in the class of strategy pairs in which the firms use the same mixed strategy over locations, and this strategy is symmetric about 0.5, there is a single equilibrium. In this equilibrium most of the probability weight of the common strategy is between 0.2 and 0.4, and between 0.6 and 0.8. There is a wide range of pure Nash (as opposed to subgame perfect) equilibrium location pairs: the subgame strategies in which each firm threatens to charge a price of zero in response to a deviation support all but those location pairs in which the firms are very close.

A Noncooperative Analysis of Hotelling's Location Game

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.

In part I of this study we established two existence theorems for mixed-strategy equilibrium in games with discontinuous payoff functions (see Theorems 4(4*) and 5(5*) in Dasgupta and Maskin (1986); hereafter D-M (1986)). In this second part we use the second of these two theorems to prove the existence of mixed-strategy equilibrium in some wellknown discontinuous economic games that fail to have pure-strategy equilibria. The models we analyse are the Bertrand-Edgeworth example of price setting duopolists with capacity constraints (see Section 2.2); price competition among firms producing differentiated products (Hotelling (1929) and d'Aspremont, Gabszewicz and Thisse (1979); see Section 2.3); spatial competition (Eaton and Lipsey (1975) and Shaked (1975); see Section 3); and models of market dependent information (Rothschild and Stiglitz (1976) and Wilson (1977); Section 4). The paper concludes (Section 5) with a classification of these models.

Today it is readily accepted that learning is in crisis and, as such, the School is in crisis; However, we believe that never before has so much been learned, which is why we will demonstrate that today the School is in crisis because education is in crisis; as such, following in the footsteps of the ancient Greeks, we will distinguish between the technique and the ethics of education, arguing that today’s School has eliminated ethics from the educational process, retaining only technique (the process of instruction). As a result, children no longer know why they should go to school and education seems purposeless; the solution to this crisis is not to perfect the education process but to recover the ethical-value dimension of education. To demonstrate this, we use the finite game/infinite game conceptual binomial developed by John Carse in Infinite and finite games.

For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.

This paper generalises the approach taken by Dasgupta & Maskin (1986) and Simon (1989) and provides necessary and sufficient conditions for the existence of pure and mixed strategy Nash equilibrium in games with continuous strategy spaces and discontinuous payoff functions. The conditions can be applied widely, and examples for existence of pure strategy and monotonic equilibria in First-Price auctions are provided. The conditions are also appropriate for ensuring that computer generated equilibrium solutions can be extended to continuous strategy spaces.

Variational convergence: Approximation and existence of equilibria in discontinuous games

Essays on Market Design and Auction Theory Youngwoo Koh This dissertation consists of three essays on market design and auction theory. In the first chapter, we develop a model of decentralized college admissions in which students’ preferences for colleges are uncertain, and colleges must incur costs when their enrollments exceed their capacities. Colleges’ admission decisions then become a tool for strategic yield management, because the enrollment at a college depends on not only students’ uncertain preferences but also other colleges’ admission decisions. We find that colleges’ equilibrium admission decisions exhibit “strategic targeting”—colleges may forgo admitting (even good) students likely sought after by the others and may admit (not as good) students likely overlooked by the others. Randomization in admissions may also emerge. The resulting assignment fails to be efficient (among students, among colleges and among all parties including colleges and students) and leads to justified envy among students. When the colleges consider multiple dimensions of students merits, their evaluations are unlikely to be perfectly correlated. In such a case, colleges may avoid head-on competition by distorting their evaluation to place excessive weight on less correlated dimensions, such as extra curricular activities and nonacademic aspects of students’ application portfolios. Restricting the number of applications or allowing for wait-listing might alleviate colleges’ yield management problem, but the resulting assignments are still inefficient and admit justified envy. Centralized matching via Gale and Shapley’s Deferred Acceptance algorithm eliminates colleges’ yield management problem and justified envy among students and attains efficiency. It also attains the outcome that is jointly optimal among colleges, but some colleges may be worse off relative to decentralized matching. The second chapter studies a keyword auction model where bidders have constrained budgets. In the absence of budget constraints, Edelman, Ostrovsky, and Schwarz (2007) and Varian (2007) analyze “locally envy-free equilibrium” or “symmetric Nash equilibrium” bidding strategies in generalized second-price (GSP) auctions. However, bidders often have to set their daily budgets when they participate in an auction; once a bidder’s payment reaches his budget, he drops out of the auction. This raises an important strategic issue that has been overlooked in the previous literature: Bidders may change their bids to inflict higher prices on their competitors because under GSP, the per-click price paid by a bidder is the next highest bid. We provide budget thresholds under which equilibria analyzed in Edelman, Ostrovsky, and Schwarz (2007) and Varian (2007) are sustained as “equilibria with budget constraints” in our setting. We then consider a simple environment with one position and two bidders and show that a search engine’s revenue with budget constraints may be larger than its revenue without budget constraints. In the third chapter, we study the procurement of an innovation in which firms exert effort and create innovations, where the quality of innovation is stochastic. Both the effort level and the quality of innovation are unverifiable, and the procurer cannot extract up-front payment from the firms. Given the uncertainty of quality realization, there is a tradeoff regarding the number of participating firms in the procurement process: If many firms participate in the process, they may be discouraged from expending their initial investment because each of them has a small chance of winning (we call this incentive effect). At the same time, as the number of participants increases, the procurer has a growing chance of getting a higher quality because of the randomness of the quality realization (sampling effect). Therefore, the procurer faces a nontrivial problem of how many firms to invite in the procurement process. We consider two prominent contest mechanisms, a first-price auction and a fixed-prize tournament. We show that if the randomness is large enough, it is optimal for the buyer to invite as many firms as possible in both mechanisms, and the fixed-prize tournament outperforms the first-price auction. In the limit at which the randomness vanishes, inviting only two firms is optimal in both mechanisms, and the firstprice auction outperforms the fixed-prize tournament. Under the first-price auction, we show that any equilibrium converges to an equilibrium as the randomness diminishes and provide a characterization of the limit equilibrium. We also provide a constructive example of a mixed-strategy equilibrium with two firms when the randomness is moderate.

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the “regular” extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models, such as the multinomial logit model or any of its variants. We show that a special equilibrium in this model has global robust stability. This means that, from any starting point, the market converges to this equilibrium when firms use a particular response mapping to dynamically adjust their own prices in response to their competitors’ prices. The mapping requires each firm to only know the demand function and cost structure for its own products (but not for other firms’ products).

This study presents a model of price competition in a market for a homogenous good with many asymmetrically positioned retailers, a typical example of which could be the online markets for books, music, movies or software. These markets are highly competitive oligopolies served by hundreds of active retailers and that have been affected by information technologies such as price comparison engines and shopping bots. In these markets, firms can only identify broad group memberships to clusters of firms, and compete within and across clusters based on this information, due to the incompleteness of information on the characteristics of firms. The characteristics of the firms that shape their competitive behavior are their established loyal segment sizes and potential comparison shopper segment sizes. To analyze such markets, we develop and solve a static game of price competition in an asymmetric oligopoly with numerous clusters of firms. In our analysis we see that the firms with the smallest loyal segment sizes and the largest switcher segment sizes engage in a fierce price competition while the members of all other groups prefer to price at their reservation price points. We also test and provide empirical support for our model predictions using pricing data on three categories. Our data set contains prices on 112 printers from 20 retailers, 95 cameras from 53 retailers and 1388 books from 15 retailers.

Les modèles de concurrence spatiale à la Hotelling permettent de traiter simultanément deux éléments de la concurrence imparfaite : la différenciation des produits et la discrimination des prix. On montre qu'au-delà d'un certain degré de monopole, la discrimination des prix permet de maximiser les résultats de la firme, notamment lorsque les coûts sont élevés et les coûts marginaux faibles. Certaines productions qui exigent des coûts fixes élevés ne peuvent alors être engagés que si une politique discriminatoire est engagée. Les entreprises ont alors intérêt à se spécialiser dans les activités où elles disposent d'un degré de monopole suffisant pour pratiquer la discrimination.

Spacing patterns in terebellid polychaetes

Currencies, competition, and clans

Why Judges Always Vote