Optimal stopping rules for multinomial observations

Many times sampling from a known or an unknown population results in observations that are integers and there is a payoff which depends on the observations. Thus, the experimenter is tempted to continue sampling in the hope of increasing his payoff. On the other hand, usually each observation costs a fixed amount and thus the experimenter also is reluctant to continue sampling because each additional sample increases the total sampling cost. The problem is to determine a place to stop making observations such that thenet payoff is a maximum. This paper considers stopping rules which will maximize the net payoff to the experimenter when the observations come from a multinomial population.     

A note on the multiple partners assignment game

In the assignment game of Shapley and Shubik [Shapley, L.S., Shubik, M., 1972. The assignment game. I. The core, International Journal of Game Theory 1, 11–130] agents are allowed to form one partnership at most. That paper proves that, in the context of firms and workers, given two stable payoffs for the firms there is a stable payoff which gives each firm the larger of the two amounts and also one which gives each of them the smaller amount. Analogous result applies to the workers. Sotomayor [Sotomayor, M., 1992. The multiple partners game. In: Majumdar, M. (Ed.), Dynamics and Equilibrium: Essays in Honor to D. Gale. Mcmillian, pp. 322–336] extends this analysis to the case where both types of agents may form more than one partnership and an agent's payoff is multi-dimensional. Instead, this note concentrates in the total payoff of the agents. It is then proved the rather unexpected result that again the maximum of any pair of stable payoffs for the firms is stable but the minimum need not be, even if we restrict the multiplicity of partnerships to one of the sides.     

Sovereign debt: Reputation, seizure and reputation

The interaction between a creditor and a sovereign debtor is described as a ‘one-shot’ game with discrete actions—total or no debt-repudiation and seizure of asset holding abroad. Possible Nash equilibria where each player chooses an action as to maximize his expected payoff given his beliefs about the other player’s action and the implications of those actions on the players’ trustworthy reputation are identified. However, if reputation losses rise convexly with the players’ relative hostility, partial repudiation and seizure can be the preferred strategies. The preferred repudiation and seizure rates are analyzed under asymmetric and symmetric information about the state of the world. (JEL classification F34)     

Organizations undertaking complex projects in uncertain environments

Recent evidence suggests that firms’ environments are becoming more complex and uncertain. This paper investigates the relationship between the complexity of a firm’s activities, environmental uncertainty and organizational structure. We assume agents are arranged hierarchically, but decisions can be made at different levels. We model a firm’s activity set as a modified NK landscape. Via simulations, we find that centralized decision making generates a higher payoff in more complex and uncertain environments, and that a flatter structure is better for the organization with centralized decision making, provided the cost of information processing is low enough. Financial Support from Zengin Foundation for Studies on Economics and Finance is gratefully acknowledged.     

Stochastic stability in assignment problems

In a dynamic model of assignment problems, it is shown that small deviations suffice to move between stable outcomes. This result is used to obtain no-selection and almost-no-selection results under the stochastic stability concept for uniform and payoff-dependent errors. There is no-selection of partner or payoff under uniform errors, nor for agents with multiple optimal partners under payoff-dependent errors. There can be selection of payoff for agents with a unique optimal partner under payoff-dependent errors. However, when every agent has a unique optimal partner, almost-no-selection is obtained.     

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.     

Learning and decision costs in one-person games

This paper reports the results of a two-part data analysis of learning in a repeated costly decision experiment. In the first part we test payoff dominance under the hypothesis of expected payoff maximization. We utilize a dynamic probability distribution over decisions for each player, characterizing what each player has learned about the payoff function by the moments from these distributions. In the second part, we test the hypothesis of expected payoff maximization for players in each treatment group. Expected payoff maximization is supported but payoff dominance is not. Copyright © 1999 John Wiley & Sons, Ltd.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Dynamic coordination among heterogeneous agents

We study a dynamic model of coordination with timing frictions and payoff heterogeneity. There is a unique equilibrium, characterized by thresholds that determine the choices of each type of agent. We characterize equilibrium for the limiting cases of vanishing timing frictions and vanishing shocks to fundamentals. A lot of conformity emerges: despite payoff heterogeneity, agents’ equilibrium thresholds partially coincide as long as a set of beliefs that would make this coincidence possible exists. However, the equilibrium thresholds never fully coincide. In case of vanishing frictions, the economy behaves almost as if all agents were equal to an average type. Conformity is not inefficient. In the efficient solution, agents follow others even more often.     

The generalized Stackelberg equilibrium of the all-pay auction with complete information

We characterize the equilibrium of the all-pay auction with general convex cost of effort and sequential effort choices. We consider a set of n players who are arbitrarily partitioned into a group of players who choose their efforts ‘early’ and a group of players who choose ‘late’. Only the player with the lowest cost of effort has a positive payoff in any equilibrium. This payoff depends on his own timing vis-a-vis the timing of others. We also show that the choice of timing can be endogenized, in which case the strongest player typically chooses ‘late’, whereas all other players are indifferent with respect to their choice of timing. In the most prominent equilibrium the player with the lowest cost of effort wins the auction at zero aggregate cost. We thank Dan Kovenock and Luis C. Corchón for discussion and helpful comments. The usual caveat applies. Wolfgang Leininger likes to express his gratitude to Wissenschaftszentrum Berlin (WZB) for its generous hospitality and financial support.     

Constrained mobility and the evolution of efficient outcomes

We study an evolutionary model akin to the one studied in Anwar (2002) where a set of agents use myopic best response learning to i) determine their action in a 2 × 2 coordination game and ii) to choose on which of multiple islands to interact. We focus on the case where the number of agents maximally allowed on each islands is constrained. We extend Anwar’s original analysis by considering the case when there may be more than two islands. We find that if the constraints are such that one island may be empty, universal coordination on the payoff dominant action is possible in the long run. If the constraints are such that all islands will be full, then for relatively mild constraints, and apart from special cases, the coexistence of conventions will occur, with one island coordinating on the risk dominant action and all remaining islands coordinating on the payoff dominant action. For relatively stringent constraints all agents will play the risk dominant action.     

Large traders and illiquid options: Hedging vs. manipulation

In this article, we study the effects on derivative pricing arising from price impacts by large traders. When a large trader issues a derivative and (partially) hedges his risk by trading in the underlying, he influences both his hedge portfolio and the derivative's payoff. In a Black–Scholes model with a price impact on the drift, we analyze the resulting trade-off by explicitly solving the utility maximization problem of a large investor endowed with an illiquid contingent claim. We find several interesting phenomena which cannot occur in frictionless markets. First, the indifference price is a convex function of the contingent claim – and not concave as in frictionless markets – implying that for any claim the buyer's indifference price is larger than the seller's indifference price. Second, the seller's indifference prices of large positions in derivatives are smaller than the Black–Scholes replication costs. Therefore, a large trader might have an incentive to issue options if they are traded at Black–Scholes prices. Furthermore, he hedges option positions only partly if he has a negative price impact and thus exploits his ability to manipulate the option's payoff. For a positive price impact he overhedges the option position leading to an extra profit from the stock position exceeding a perfect hedge. Finally, we also study a model where the large shareholder has a price impact on both drift and volatility.     

Valuing lookback options with barrier

In this paper, we introduce a new class of exotic options, termed lookback-barrier options, which literally combine lookback and barrier options by incorporating an activating barrier condition into the European lookback payoff. A prototype of lookback-barrier option was first proposed by Bermin (1998), where he intended to reduce the expensive cost of lookback option by considering lookback options with barrier. However, despite his novel trial, it has not attracted much attention yet. Thus, in this paper, we revisit the idea and extend the horizon of lookback-barrier option in order to enhance the marketability and applicability to equity-linked investments. Devising a variety of payoffs, this paper develops a complete valuation framework which allows for closed-form pricing formulas under the Black–Scholes model. Our closed-form pricing formulas provide a substantial advantage over the method of Monte Carlo simulation, because the extrema appearing in both of the lookback payoff and barrier condition would require a large number of simulations for exact calculation. Complexities involved in the derivation process would be resolved by the Esscher transform and the reflection principle of the Brownian motion. We illustrate our results with numerical examples.     

非常规项目下内部收益率修正方法的比较分析

在项目评估过程中,对项目盈利能力评价的一个重要指标是内部收益率,但内部收益率在非常规项目评估中却不能有效的发挥作用,本文通过对内部收益率的修正方法进行比较分析,提出修正内部收益率是其中比较实用的方法。     

On optimality of illegal collusion in contracts

Illegal collusion is a widespread phenomenon all around the world. Yet, models of hierarchical agency relationships tend not to predict collusion. This paper demonstrates that a natural requirement of interim efficiency suffices for collusion to appear in equilibrium in a simple standard setting. The optimal extent of collusion depends on the efficacy of the legal system. When the transaction costs associated with illegal deals are small enough, inducing some illegal collusion between the agent and his supervisor increases the principal's payoff. Received: 9 December 1996 / Accepted: 11 April 1998     

Private Information and Bargaining Power in Venture Capital Financing

We model the natural evolution of private information over the life of a venture capitalist financed project. In the early stages, the entrepreneur is better informed regarding the project, and when the project matures, the venture capitalist has an informational advantage over the entrepreneur. Within this framework, we examine how the venture capitalist's relative bargaining power affects cash flow rights and investment. When the bargaining advantage lies with the entrepreneur, the project may not be screened, and the venture capitalist may acquiesce to excessive initial investment but subsequently terminate the project. Increased venture capitalist bargaining power encourages project screening, attenuates the incentive to overinvest, and reduces the incidence of project termination subsequent to the initial investment. The payoff sensitivity of venture capitalist's financing contract also increases as his bargaining power improves.     

First price package auction with many traders

This paper studies a first price package auction in which multiple sellers participate in addition to multiple buyers. We generalize the notion of the profit-target strategy which is first introduced as a truthful strategy in a first price package auction with a single seller by Bernheim and Whinston (1986b). We then show that the set of equilibrium payoffs in profit-target strategies is equal to the bidder-optimal core, and is also equal to the set of coalition-proof Nash equilibria. Using this result, we find that any equilibrium payoff vector is weakly Pareto-dominated by the VCG payoff vector for buyers, and that the Walrasian competitive equilibrium payoff vector is weakly Pareto-dominated by some equilibrium payoff vector for buyers, even if goods are substitutes. This contrasts with the first price package auction with a single seller, in which it is shown that if goods are substitutes, then those three outcomes are payoff-equivalent.     

Solving dynamic public insurance games with endogenous agent distributions: Theory and computational approximation

We make two contributions in this paper. First, we extend the characterization of equilibrium payoff correspondences in history-dependent dynamic policy games to a class with endogenously heterogeneous private agents. In contrast to policy games involving representative agents, this extension has interesting consequences as it implies additional nonlinearity (i.e., bilinearity) between the game states (distributions) and continuation/promised values in the policymaker’s objective and incentive constraints. The second contribution of our paper is in addressing the computational challenges arising from this payoff-relevant nonlinearity. Exploiting the game’s structure, we propose implementable approximate bilinear programming formulations to construct estimates of the equilibrium value correspondence. Our approximation method respects the property of upper hemicontinuity in the target correspondence. We provide small-scale computational examples as proofs of concept.     

Applying cooperative game theory to power relations

The power of people and the power of some people over others have long been a topic in sociology, politics, psychology, and, to a lesser extent, economics. This paper shows how cooperative game theory can be used to analyze the power relation between agents 1 and 2. First, we identify payoff reflections of the power of agent 1 over agent 2. Second, we suggest how to look for action reflections of power-over. We will provide many examples and reconsider (an important part of) the power-over literature from the viewpoint of our approach.     

Observing each other’s observations in a Bayesian coordination game

We study a Bayesian coordination game where agents receive private information on the game’s payoff structure. In addition, agents receive private signals that inform them of each other’s private information. We show, that once agents possess these different types of information, there exists a coordination game in the evaluation of this information. Even though the precisions of both signal types are exogenous, the precision with which agents forecast each other’s actions in equilibrium turns out to be endogenous. As a consequence, there exist multiple equilibria which differ with regard to the way that agents weight their private information to forecast each other’s actions.