Todd R. Kaplan

A Program for Finding Nash Equilibria

We describe two-player simultaneous-play games. First, we use a zero-sum game to illustrate minimax, dominant, and best-response strategies. We illustrate Nash equilibria in the Prisoners’ Dilemma and the Battle of the Sexes Games, distinguishing among three types of Nash equilibria: a pure strategy, a mixed strategy, and a continuum (partially) mixed strategy. Then we introduce the program, Nash . m, and use it to solve sample games. We display the full code of Nash . m; finally, we discuss the performance characteristics of Nash . m.

All-Pay Auctions with Variable Rewards

We study all-pay auctions with variable rewards under incomplete information. In standard models, a reward depends on a bidders privately known type; however, in our model it is also a function of his bid. We show that in such models there is a potential for paradoxical behavior where a reduction in the rewards or an increase in costs may increase the expected sum of bids or alternatively the expected highest bid. Copyright 2002 by Blackwell Publishing Ltd

Innovative Activity and Sunk Cost

We analyze a patent race where the first innovator receives a time-dependent reward while all firms incur costs. When firms are identical, there is a unique, symmetric, mixed-strategy equilibrium that yields zero expected profits for all firms. Furthermore, the expected innovation time is an increasing function of the number of firms and a decreasing function of the size of the reward. When one firm has a higher reward than another, it is more likely to win. Although similar to an all-pay auction, our approach may yield both similar and qualitatively different behavior.

Influence of aggregation and measurement scale on ranking a compromise alternative in AHP

Analytic Hierarchy Process (AHP) is one of the most popular multi-attribute decision aid methods. However, within AHP, there are several competing preference measurement scales and aggregation techniques. In this paper, we compare these possibilities using a decision problem with an inherent trade-off between two criteria. A decision-maker has to choose among three alternatives: two extremes and one compromise. Six different measurement scales described previously in the literature and the new proposed logarithmic scale are considered for applying the additive and the multiplicative aggregation techniques. The results are compared with the standard consumer choice theory. We find that with the geometric and power scales a compromise is never selected when aggregation is additive and rarely when aggregation is multiplicative, while the logarithmic scale used with the multiplicative aggregation most often selects the compromise that is desirable by consumer choice theory.

Does AHP Help Us Make a Choice? An Experimental Evaluation

In this paper, we use experimental economics methods to test how well Analytic Hierarchy Process (AHP) fares as a choice support system in a real decision problem. AHP provides a ranking that we statistically compare with three additional rankings given by the subjects in the experiment: one at the beginning, one after providing AHP with the necessary pair-wise comparisons and one after learning the ranking provided by AHP. While the rankings vary widely across subjects, we observe that for each individual all four rankings are similar. Hence, subjects are consistent and AHP is, for the most part, able to replicate their rankings. Furthermore, while the rankings are similar, we do find that the AHP ranking helps the decision makers reformulate their choices by taking into account suggestions made by AHP.

Dividing the indivisible: procedures for allocating cabinet ministries to political parties in a parliamentary system

Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each other’s preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is non-monotonicity – choosing earlier may hurt rather than help a party. We introduce a mechanism, combining sequential choices with a structured form of trading, that results in sincere choices for two parties that avoids these problems. Although there are dif.culties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to make consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.

The Strategic Use of Seller Information in Private-Value Auctions

In the framework of a first-price private-value auction, we study the seller as a player in a game with the buyers in which he has private information about their realized valuations. We find that depending upon his information, set of signals, and commitment power, he may strategically transmit messages to buyers in order to increase his revenue. In an environment where the seller knows the rankings and lacks any commitment power, we find that the seller is unable to exploit his information. However, in an environment where the seller knows the realized valuations and can credibly announce either the true rankings or the true values (or announce nothing at all) but cannot commit as to which of these truthful messages to announce, then it is indeed possible to increase his revenue. If the seller, in addition, can commit to the full signaling strategy, then his expected revenue will be even higher. We believe that this line of research is fruitful for both better understanding behavior in auctions and finding paths to higher seller revenue.

Which Way to Cooperate

We introduce a two-player, binary-choice game in which both players have a privately known incentive to enter, yet the combined surplus is highest if only one enters. Repetition of this game admits two distinct ways to cooperate: turn taking and cutoffs, which rely on the players private value to entry. A series of experiments highlights the role of private information in determining which mode players adopt. If an individuals entry values vary little (e.g., mundane tasks), taking turns is likely; if these potential values are diverse (e.g., difficult tasks that differentiate individuals by skill or preferences), cutoff cooperation emerges.

Effective price-matching: a comment

Abstract Corts [Economic Lett. 47 (1995) 417] showed that when allowing for price-beating policies in addition to price-matching policies, the competitive outcome prevails in lieu of monopoly pricing. I show by expanding the strategy set further to include effective price strategies, the possibility of monopoly pricing is restored.

Second-Price Auctions with Private Entry Costs

We study asymmetric second-price auctions under incomplete information. The bidders have potentially different, commonly-known, valuations for the object and private information about their entry costs. The seller, however, does not benefit from these entry costs. We calculate the equilibrium strategies of the bidders and analyze the optimal design for the seller in this environment.