H. PEYTON YOUNG

Scott and Barbara Black Professor of Economics
Johns Hopkins University, USA
Professor of Economics
University of Oxford, UK
Senior Fellow
The Brookings Institution, USA
 


Excerpt from H. Peyton Young's interview: Question 4 & 5. What do you consider the most neglected topics and/or contributions in late 20th century game theory? What are the most important open problems in game theory and what are  the prospects for progress?

I will address these last two questions in tandem.  As I mentioned earlier, cooperative game theory is an unjustly neglected topic of research .  This was not always the case:  von Neumann and Morgenstern put a great deal of emphasis on the cooperative form, and many of the pioneers in game theory made major contributions to the topic (Shapley, 1953; Aumann and Maschler, 1964; Schmeidler, 1969; Aumann and Shapley, 1974).   In recent decades, however, the noncooperative approach has increasingly gained the upper hand.  Indeed, this trend has gone so far that many textbooks on game theory scarcely give cooperative theory a mention.   One reason for this development, as I have already suggested, is that the topics in economics where game theory made its earliest inroads -- mechanism design and industrial organization – seem particularly well-suited to the noncooperative approach. 

Another reason why cooperative game theory has languished is that its practical applications have not been widely recognized. Earlier I mentioned the problem of sharing costs among the beneficiaries of a public facility. Similar problems arise in setting rates for public utilities (Zajac, 1978).  More generally, cooperative game theory is relevant to any situation where scarce resources are to be allocated fairly among a group of claimants.  How, for example, should slots at busy airports be allocated among airlines?  Which transplant patient should be first in line for the next kidney?   How should political representation in a national legislature be fairly divided among parties and geographical regions?   Some economists insist that such problems would be solved if they were simply left to the workings of the market.  Unfortunately, this overlooks the point that markets are moot unless property rights have been defined and vested in individuals, which is precisely what methods of fair allocation are about.  

In my book, Equity In Theory and Practice (1994), I examined various fairness concepts from both a foundational and practical standpoint. Cooperative solution concepts like the core and the Shapley value, as well as semi-cooperative notions like the Nash bargaining solution and the Kalai-Smorodinsky solution, provide the entry point for thinking about the meaning of allocative fairness.   A close examination of practice, however, suggests that one must go substantially beyond these approaches to formulate a theory that has descriptive validity. 

Three central points emerge from the analysis.  First, fairness must be judged in the context of the problem at hand.   Criteria for allotting transplant organs may be quite different from criteria that pertain to the allocation of legislative seats, and neither may be relevant to the allocation of offices in the workplace or dormitory rooms at college.  In other words, notions of justice tend to be compartmentalized and context-specific, a view that has its roots in Aristotelian philosophy, and has been advanced by political philosophers such as Walzer (1983) and Elster (1992).    

A second key point is that, in practice, solutions to fairness problems tend to be decentralized in the following sense: an allocation is deemed to be fair for a group of claimants only when every subgroup deems that they fairly divide the resources allotted to them. This subgroup consistency principle is very ancient. It is implicit, for example, in certain Talmudic doctrines concerning the division of inheritances (Aumann and Maschler, 1985).  It also features in many modern solution concepts, such as the core, the nucleolus, and the Nash bargaining solution (Sobolev, 1975; Lensberg, 1988), and in real-world allocation methods such as rules for apportioning seats in legislatures (Balinski and Young, 1982; Young, 1994). 

The cooperative game approach to fair division proceeds from an axiomatic standpoint.   There is, however, another way of thinking about fairness norms that builds on noncooperative game theory.  Norms of fair division – indeed norms in general – are often the unpremeditated outcome of historical chance and precedent.   What is fair in one society may not be deemed fair in another, because people’s expectations are conditioned by precedent, and precedents accumulate through the vagaries of history.  

Such processes can be modeled noncooperatively using the framework of evolutionary game theory.   As I mentioned earlier, this approach was originally inspired by biological applications, and typically has three key features: i) there is a large population of interacting players; ii) the players have heterogeneous characteristics, including different payoffs, information, and behavioral repertoires, iii) they adapt their behavior based on local conditions and experience, and are purposeful but not always perfectly rational.  The focus is on the dynamics of such a process, not merely on its equilibrium states.  One of the main contributions of the theory is to show that some equilibria have a much higher probability of arising than do others (Foster and Young, 1990; Kandori, Mailath, and Rob, 1993; Young, 1993a).   It therefore delivers a theory of equilibrium selection that is based on evolutionary principles rather than on a priori principles of ‘reasonableness’, as in the earlier theory developed by Harsanyi and Selten (1988). 

To illustrate how the evolutionary approach can be applied to the study of fairness norms, consider the classical problem of how two individuals would divide a pie.    The simplest noncooperative formulation is due to John Nash (1950): each player names a fraction of the pie, and they get their demands provided that both can be satisfied; otherwise they get nothing.   Any pair of demands that sums to unity constitutes a noncooperative equilibrium of the one-shot game.  If the players are allowed to bargain over time, much tighter predictions are possible. In the standard model, players alternate in making demands, which are either accepted or rejected (Stahl, 1972; Rubinstein, 1982).  When the players are perfectly rational and discount future payoffs at the same rate, the outcome of the unique subgame perfect equilibrium is the Nash bargaining solution.   

Neither the one-shot demand game nor the alternating offers game is evolutionary in spirit, because they are concerned with what two particular bargainers would do in equilibrium, not what a population of bargainers would do.  To recast the problem in an evolutionary framework, consider a large population of agents who engage in pairwise bargains from time to time. Suppose that the outcomes of previous bargains affect how people bargain in the future, due to the salience of precedent.  Once a particular way of dividing the pie becomes entrenched due to custom, people start to think that this is the only fair and proper way to divide the pie, and it therefore continues in force.   

To allow for asymmetric interactions, suppose that there are two distinct populations of potential bargainers who are randomly matched each period (e.g., employers and employees).  Each matched pair plays the Nash demand game described earlier.  Assume for simplicity that all agents in a given population  have the same utility function, but that the utility functions differ between populations. To capture the idea that current expectations are shaped by precedent, suppose that each current player looks at a random sample of earlier demands by the opposing side, and chooses a trembled best reply given the sample frequency distribution.  (The ‘tremble’ captures the idea that the process is jostled by small unobserved utility shocks, so that players usually choose a best reply but not always.)  It can be shown that, starting from arbitrary initial conditions, players’ expectations eventually coalesce around a specific division of the pie, and this endogenously generated norm of division is, with high probability, the Nash bargaining solution.  Furthermore, when players are heterogeneous with respect to their degree of risk aversion, a natural generalization of the Nash bargaining solution results (Young, 1993b). 

This example shows that there is no need to make extreme assumptions about players’ rationality in order for game theory to yield interesting results. Unlike the alternating offers model, where perfect rationality and common knowledge of perfect rationality are assumed, neither is needed in the evolutionary model. Players choose myopic best replies based on fragmentary information, they occasionally make mistakes, and they have no a priori knowledge of their opponents’ payoffs, behaviors, or degree of rationality.  Nevertheless the two models yield essentially the same outcome. 

More generally, the evolutionary model of bargaining illustrates how game theory can be used to study the emergence of norms. Over time, interactions among people build up a stock of precedents that may cause their expectations to gravitate toward a particular equilibrium, which then becomes entrenched as a social norm: everyone adheres to it because everyone expects everyone else to adhere to it.  When the underlying game is concerned with the division of scarce resources, the resulting equilibrium can be interpreted as a fairness norm (Hume, 1739;  Binmore, 1994;  Young , 1998).

I conclude by hazarding several predictions about the future development of game theory. The first is that rationality, and arguments over how rational  the players “really” are, will fade in importance.  As I have already argued, game theory can be applied to systems of interacting agents whether or not they are rational in the conventional sense.  This insight was initially provided by applications of game theory to biology, and is being buttressed by current applications to computer science, artificial intelligence, and distributed learning.

 My second prediction is that game theory will continue to evolve in response to real problems that arise in economics, politics, computing, philosophy, biology and other subjects, a development that von Neumann and Morgenstern would surely have welcomed.  While its major successes to date have largely been in economics, game theory is not a sub-discipline of economics; it is more like statistics, a subject in its own right with applications across the academic spectrum.  

 My third prediction is more of an admonition: game theory will continue to thrive if it remains receptive to new ideas suggested by applications, but risks  degenerating if it does not.  John von Neumann cautioned about this tendency in mathematics more generally, and game theorists would do well to heed his warning (von Neumann, 1956): 

 “I think that it is a relatively good approximation to truth – which is much too complicated to allow anything but approximations – that mathematical ideas originate in empirics… As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality,” it is beset with very grave dangers.  It becomes more and more purely aestheticizing, more and more purely l’art pour l’art.  … [W]henever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.”

Read the remaining part of Peyton Young's interview in Game Theory: 5 Questions, edited by Vincent F. Hendricks and Pelle Guldborg Hansen. The book is released in April 2007 by Automatic Press / VIP.

ISBN 87-991013-4-3
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