Topological games at Princeton, a mathematical memoir

The games of the title are “Nash” (or Hex), “Milnor” (or Y), “Shapley” (or Projective Plane) and “Gale” (or Bridg-It) all of which were discovered (or re-discovered) in Princeton in 1948–1949. After giving the basic topological connections, I will discuss more recent ramifications related to computational complexity theory. A recurrent theme will be non-constructive proofs, or how we can know something can be done without having the slightest idea of how to do it.     

Contracts and externalities: How things fall apart

A single principal interacts with several agents, offering them contracts. The crucial assumption of this paper is that the outside-option payoffs of the agents depend positively on how many uncontracted or “free” agents there are. We study how such a principal, unwelcome though he may be, approaches the problem of contract provision to agents when coordination failure among the latter group is explicitly ruled out. Two variants are considered. When the principal cannot re-approach agents, there is a unique equilibrium, in which contract provision is split up into two phases. In phase 1, simultaneous offers at good (though varying) terms are made to a number of agents. In phase 2, offers must be made sequentially, and their values are “discontinuously” lower: they are close to the very lowest of all the outside options. When the principal can repeatedly approach the same agent, there is a multiplicity of equilibria. In some of these, the agents have the power to force delay. They can hold off the principal's overtures temporarily, but they must succumb in finite time. In both models, despite being able to coordinate their actions, agents cannot resist an “invasion” by the principal and hold to their best payoff. It is in this sense that “things [eventually] fall apart”.     

Sequential Equilibrium and Competition in a Diamond–Dybvig Banking Model

Within the framework of a Diamond–Dybvig model [J. Polit. Econ.91(1983), 401–419], but with explicitly modelling the autarky choice during the planning period, we demonstrate that a mixed strategy bank run equilibrium that does not rely on sunspots may coexist with the sunspot run equilibrium previously studied in the literature. In a version of the model with multiple banks, there exist sequential equilibria that imply positive profits. However, the zero-profit contract in which runs never occur can be supported as the unique equilibrium outcome if the agents play pure strategies only and their beliefs are restricted to be consistennt with a forward induction argument.Journal of Economic LiteratureClassification Numbers: C72, G21     

On a probabilistic model of forecasting

One of the methods of studying complex objects is the construction of a mathematical model, containing such information about the object that is necessary to solve a definite problem connected with it.Mathematical modeling, based on the construction of models of various kinds can be used in forecasting. Let a forecasting object A(X) be described by vector X = (X₁, X₂,…,X_n) whose coordinates are parameters characterizing this object. The work presents a probabilistic model of forecasting and gives the example of a forecast of the object described by a set two parameters.     

Capturing the pharmaceutical value of biodiversity in a developing country

The conservation of biodiversity is increasingly being defined as a process made up of three components—saving, studying and sustainably using biodiversity. If the conservation of process is expanded to incorporate the need for both the protection of biodiversity—either in situ or ex situ—and the generation of valuable information about biodiversity, an important issue is the correct balance of investment in these two types of activities. This paper explores this issue by focusing on efforts in a developing country to capture the pharmaceutical potential of biodiversity and species information. A model of biodiversity investment choice is developed, and an analysis of royalty returns to effort in biodiversity protection, taxonomic information and the collection of biotic samples is conducted based on data from Costa Rica. The results suggest that although a country can be adequately compensated for its investment in the generation of taxonomic information, it is unlikely that pharmaceutical prospecting alone will cover the high opportunity costs of biodiversity protection.     

Consistent representative democracy

We study axioms which define “representative democracy” in an environment in which agents vote over a finite set of alternatives. We focus on a property that states that whether votes are aggregated directly or indirectly makes no difference. We call this property representative consistency. Representative consistency formalizes the idea that a voting rule should be immune to gerrymandering. We characterize the class of rules satisfying unanimity, anonymity, and representative consistency. We call these rules “partial priority rules.” A partial priority rule can be interpreted as a rule in which each agent can “veto” certain alternatives. We investigate the implications of imposing other axioms to the list specified above. We also study the partial priority rules in the context of specific economic models.     

University education in future studies— a mark III critique: Part II

In 1970 an initial survey was made of futures courses at university level based on forty courses that could be located in the United States and Canada [Technol. Forecast. Soc. Change 2, 133–148 (1970)]. This was replicated during 1970–1972 on approximately 200 North American courses and reported changes in the field at the Third World Future Research Conference in Bucharest, September 1972 (Technol. Forecast. Soc. Change, 4, 387–407 [1973]). A third interation was funded by the World Institute Council and published in abstracted form with their permission here. The conclusions to be drawn from some six years of analysis of such courses, and drawn especially from the specifics of the 300 courses in the third iteration, can be lumped under five main headings: (1) All disciplines are to some degree finally alerted to the future implications of both their research and their teaching. (2) There remains still much “froth and nonsense” in futures courses. How to control this and encourage lively creativity is a complex problem-especially since the intellectual image of futurism today is not that high! (3) There appears to be no reliable sociocultural change theory backing future studies and there is precious little solid theory backing future studies and there is precious little solid theory in the field itself-offset by some progress in methodology. (4) Despite these strictures, if all future-oriented courses in American/Canadian universities were lumped together (future Studies per se; technology forecasting/assessment; policy sciences; peace studies; demography; environmental/ecology courses; general systems theory/system analysis/system dynamics; mathematical modelling and game theory; eschatology; utopian literature and science fiction; even black studies and women's courses) the total might approach 5000 and it is still growing, as is popular and academic understanding of the necessity “to plan ahead” in this period of human malaise. (5) Informal education of varied worth, from free swinging communes to middle management cram courses conducted by think-tanks, may turn out to be more useful in future-oriented advanced education than that which takes place in formal “walled” traditional institutions.     

“Constitutionalizing” the corporation

To become effective instruments in the attainment of a sustainable society, corporations should have their constitutional status settled. Thus far, mainly by Supreme Court decisions, corporations have an uncertain status. They are “persons” under the Constitution and have certain rights; but unlike natural persons, they do not have concomitant constitutional duties. Corporations should be viewed as “private” governments exercising substantial power in society. They are, however, considered to be associations of individuals rather than divisions of society. The need is to legitimate their governing power by “constitutionalizing” them. That can be done by corporations accepting, or having imposed upon them, two principal constitutional duties. First, means should be developed whereby corporate officers routinely take the general good into account. That duty could be implemented by making “social impact statements,” analogous to but broader than the familiar environmental impact statements, before making important corporate decisions. Secondly, as “sociological communities,” corporate officers wield considerable power over members of those communities. A bill of rights similar to the Constitution's Bill of Rights is recommended so as to make corporate power that is necessary for achievement of societal goals as tolerable and decent as possible. Acceptance of those duties would make corporations parts of, not separate from, the greater corporation called society. They would be private, profit-making entities with a definite public function. Preferably, the duties should be voluntarily accepted. Congress, however, has undoubted constitutional power to impose both, should it so desire.     

The Core of Large Differentiable TU Games

For suitable non-atomic TU games ν, the core can be determined by computing appropriate derivatives of ν, yielding one of two stark conclusions: either core(ν) is empty or it consists of a single measure that can be expressed explicitly in terms of derivatives of ν. In this sense, core theory for a class of games may be reduced to calculus. Journal of Economic Literature Classification Number: C71.     

On the Existence of Stable Roommate Matchings

This paper identifies a condition called “no odd rings” that is sufficient for the existence of stable roommate matchings in the weak preferences case. It shows that the process of allowing randomly chosen blocking pairs to match converges to a stable roommate matching with probability one as long as there are no odd rings. This random-paths-to-stability result generalizes that of Roth and Vande Vate (1990, Econometrica58, 1475–1480) and may not hold if there are odd rings. The “no odd rings” condition can also be used to prove a number of other sufficient conditions that are more economically interpretable. Journal of Economic Literature Classification Numbers: C78, D71.     

Hedging and flexing : Two ways of choosing under ignorance

Two strategies for making decisions under severe uncertainty—called here ignorance—are contrasted. A popular strategy under such conditions is to forego the best outcome in order to avoid the worst which might happen; this is hedging, and is shown to lead to a number of problems: once made, a hedge can rarely be revised if the worst does not occur; hedging often brings about the worst outcome which is being hedged against; the worst outcome is often undefined; and hedging can lead to a vicious circle, disregarding the benefits which an option might bring. Each problem is illustrated by a decision about technological change, which typically are decisions under ignorance. Flexing has none of these problems. Flexing aims at achieving the very best outcome. This too risks achieving the worst outcome, but the strategy insures against this. The decision maker should ensure that if the worst outcome arises, the original decision can be reserved quickly and without great cost. Ways in which this can be done are discussed through the examples.     

Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views

We provide a new interpretation of mixed strategy equilibria that incorporates both von Neumann and Morgenstern's classical concealment role of mixing, as well as the more recent Bayesian view originating with Harsanyi. For any two-person game, G, we consider an incomplete information game, in which each player's type is the probability he assigns to the event that his mixed strategy in G is “found out” by his opponent. We show that, generically, any regular equilibrium of G can be approximated by an equilibrium of in which almost every type of each player is strictly optimizing. This leads us to interpret i's equilibrium mixed strategy in G as a combination of deliberate randomization by i together with uncertainty on j's part about which randomization i will employ. We also show that such randomization is not unusual: for example, i's randomization is nondegenerate whenever the support of an equilibrium contains cyclic best replies.     

Contests with a stochastic number of players

We study Tullock's (1980) n-player contest when each player has an independent probability 0 < p 1 of participating. A unique symmetric equilibrium is found for any n and p and its properties are analyzed. In particular, we show that for a fixed n > 2 individual equilibrium spending as a function of p is single-peaked and satisfies a single-crossing property for any two different numbers of potential players. However, total equilibrium spending is monotonically increasing in p and n. We also demonstrate that ex-post over-dissipation is a feature of the pure-strategy equilibrium in our model. It turns out that if the contest designer can strategically decide whether to reveal the actual number of participating players or not, then the actual number of participants is always revealed.     

Dynamic stability and reform of political institutions

This paper examines endogenous institutional change in a class of dynamic political games. The political aggregation rules used at date t+1 are instrumental choices under rules at date t. Effectively, rules are “players” who can strategically delegate future policy-making authority to different rules. A political rule is stable if it selects itself. A reform occurs when an alternative rule is selected. The stability of a political rule is shown to depend on whether its choices are dynamically consistent. For instance, simple majority rules can be shown to be dynamically consistent in many common environments where wealth-weighted voting rules are not. The result extends to political rules that incorporate private activities such as extra-legal protests, threats, or private investment. The approach is one way of understanding various explanations of institutional change proposed in the literature. A parametric model of public goods provision gives an illustration.     

Reference Points and Negative Reciprocity in Simple Sequential Games

We investigate experimentally whether preferences over an outcome depend on what other possible outcomes of the situation under consideration are, i.e., whether choices are “menu dependent.” In simple sequential games we analyze whether reactions to a certain benchmark outcome are influenced by changes in the payoffs of another outcome, not attainable at that time, called the “reference point.” Our data provide evidence that is favorable to the notion of menu dependence. Alterations of the reference point can lead to quantitatively significant changes in behavior at the benchmark outcome. The behavior we observe can be interpreted in terms of negative reciprocity. Journal of Economic Literature Classification Numbers: C91, C92, C72.     

Imitation processes with small mutations

This note characterizes the impact of adding rare stochastic mutations to an “imitation dynamic,” meaning a process with the properties that absent strategies remain absent, and non-homogeneous states are transient. The resulting system will spend almost all of its time at the absorbing states of the no-mutation process. The work of Freidlin and Wentzell [Random Perturbations of Dynamical Systems, Springer, New York, 1984] and its extensions provide a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply. This note provides a simpler and more intuitive algorithm. Loosely speaking, in a process with K strategies, it is sufficient to find the invariant distribution of a K×K Markov matrix on the K homogeneous states, where the probability of a transit from “all play i” to “all play j” is the probability of a transition from the state “all agents but 1 play i, 1 plays j” to the state “all play j”.     

Forecasting and planning in an incoherent context

What follows concerning the predicament of forecasts and planning is subject to the same criticisms and constraints I apply to them and their context. My observations are also forecasts: stories based on arbitrary and fragmentary images of social “reality” and my words share the same dubious status as words per se that I shall describe later. But all of us are sinking in this ontological and epistemological swamp: To paraphrase the Tao Te Ching, those who know can not say: those who say do not know. So be it.     

Decentralized decision-making in a large team with local information

We study a problem involving a team of agents each associated with a node in a chain. Each agent makes a decision that influences only his own cost and those of adjacent agents. Prior to making his decision, each agent observes only the cost structure associated with nodes that can be reached by traversing no more than r arcs. Decisions are selected without any coordination, with the common objective of minimizing average cost among agents. We consider such decisions decentralized since agents act based on different information. Cost incurred by an optimal centralized strategy, in which a single decision-maker has access to all information and dictates all decisions, is employed as a performance benchmark. We show that, to maintain a certain level of performance relative to optimal centralized strategies, decentralized deterministic strategies require r to be proportional to the number of agents. This means that the amount of information accessible to any agent should be proportional to the total number of agents. Stochastic strategies, on the other hand, decentralize more gracefully—the amount of information required by each agent is independent of the total number of agents.     

Aggregation of semiorders: intransitive indifference makes a difference

Summary A semiorder can be thought of as a binary relationP for which there is a utilityu representing it in the following sense:xPy iffu(x) –u(y) > 1. We argue that weak orders (for which indifference is transitive) can not be considered a successful approximation of semiorders; for instance, a utility function representing a semiorder in the manner mentioned above is almost unique, i.e. cardinal and not only ordinal. In this paper we deal with semiorders on a product space and their relation to given semiorders on the original spaces. Following the intuition of Rubinstein we find surprising results: with the appropriate framework, it turns out that a Savage-type expected utility requires significantly weaker axioms than it does in the context of weak orders.We wish to thank Tatsuro Ichiishi, Jorge Nieto, Ariel Rubinstein, Efraim Sadka and especially David Schmeidler and anonymous referees for stimulating discussions and comments. I. Gilboa received partial financial support from NSF grants nos. IRI-8814672 and SES-9113108, as well as from the Alfred P. Sloan Foundation.