## Game Theory and Strategic Interaction

## Game Theory and Strategic Interaction

# GAME THEORY AND STRATEGIC INTERACTION

A *game* is a situation that involves two or more decision makers (called *players*), where (1) each player faces a choice between at least two behavioral options, (2) each player strives to maximize utility (i.e., to achieve the greatest payoff possible), and (3) the payoff obtained by a given player depends not only on the option that he or she chooses but also on the option(s) chosen by the other player(s). In virtually all games, some or all of the players have fully or partially opposing interests; this causes the behavior of players to be proactive and strategic.

The *theory of games* is a branch of applied mathematics that rigorously treats the topic of optimal behavior in two-person and *n*-person games. Its origins go back at least to 1710, when the German mathematician-philosopher Leibniz foresaw the need for a theory of games of strategy. Soon afterward, James Waldegrave (in Montmort 1713; 1980) formulated the concept of maximin, a decision criterion important to game theory. In his book *Mathematical Psychics*, Edgeworth (1881; 1995) made explicit the similarity between economic processes and games of strategy. Later, theorists such as Zermelo (1913) stated specialized propositions for certain games (e.g., chess). Not until the work of Borel ([1921–1927] 1953) and von Neumann (1928), however, did the foundations of a true theory of games appear. A landmark of the modern era, von Neumann and Morgenstern's *Theory of Games and Economic Behavior* (1944), extended game theory to problems involving more than two players. Luce and Raiffa (1957) published the first widely used textbook in game theory. For more details regarding the early history of game theory, see Dimand and Dimand (1996) and Weintraub (1992).

Game theory has continued to develop substantially in recent years. Many introductory presentations of the modern mathematical theory are available. Among these are Friedman (1990), Jones (1980), Myerson (1991), Owen (1982), Romp (1997), and Szep and Forgo (1985).

Beyond its status as a branch of applied mathematics, game theory serves social scientists as a tool for studying situations and institutions with multiple decision makers. Some of these investigations are empirical, while others are primarily analytic in character. The dependent variables of central concern in games include allocation of payoffs (i.e., who receives what rewards or bears what costs) and formation of coalitions (i.e., which of various possible alliances among players occur in a game.) Other concerns include whether outcomes of a game are stable or not, whether outcomes are collectively efficient or not, and whether outcomes are fair or not in some specific sense.

## GAME-THEORETIC CONCEPTS

Mathematical game theory provides three main tools that assist in the analysis of multiperson decision problems. These include a descriptive framework, a typology of games, and a variety of solution concepts.

**Descriptive Framework.** At base, a description of any game requires a list of all players, the strategies available to each player, the logically possible outcomes in the game, and the payoff of each outcome to each player. In some instances, a game's description also includes a specification of the dynamic sequence of play and of the (possibly limited or incomplete) information sets available to players. Payoffs in a game are expressed in terms of *utility*; this provides a standard means of comparing otherwise diverse outcomes.

An analyst can model or represent a game in various forms. *Extensive form* depicts all possible strategies of players in a tree format. It is especially useful for modeling games in which play occurs in stages or over time. *Strategic form* (also called normal form or a "payoff matrix") shows payoffs to players as a function of all strategy combinations. *Characteristic function form* lists the minimum payoffs assured for each of the coalitions in a game. Whereas extensive and normal forms pertain to virtually all types of games, characteristic function form pertains only to cooperative games (i.e., games that permit coalitions).

**Typology of Games.** The second tool from game theory is a general typology of games. This provides a means of codifying or classifying games vis-à-vis one another. At base, there are four major types of games. Games can be either *static* (i.e., single time period) or *dynamic* (multiple time periods), and they can involve either *complete information* (all relevant information is shared and held in common) or *incomplete information* (some information is private and held only by some players). Much of classic game theory was formulated with reference to static games involving complete information; more recent developments have extended the theory to dynamic games and also to games involving incomplete information.

Games can be *two-person* or *n-person* (more than two players), and they can be further classified as *cooperative* or *noncooperative*. Cooperative games permit players to communicate before reaching decisions and include some mechanism that enables players to make binding agreements regarding coordination of strategies. Noncooperative games do not permit players to communicate or to form binding agreements prior to play. In other words, cooperative games enable players to form coalitions whereas noncooperative games do not.

Among cooperative games, some are *sidepayment games* while others are *nonsidepayment games*. Sidepayment games permit players to transfer payoffs (utility) within coalitions; nonsidepayment games do not. A further distinction applicable to cooperative sidepayment games is that between *simple games* and *nonsimple games*. Simple games are those in which the characteristic function assumes only two values, whereas nonsimple games are those in which the characteristic function has more than two values. Analysts use simple games primarily to model social processes with binary outcomes (e.g., win–lose, succeed–fail, etc.)

**Solution Concepts.** The third set of tools provided by game theory is a variety of solution concepts. A *solution concept* is theory of equilibrium that predicts (behaviorally) or prescribes (normatively) the allocation of payoffs to players in games. In other words, a solution concept specifies how a game will turn out when played. For this reason, solution concepts are among the most important contributions of game theory.

Game theorists have developed numerous solution concepts. These differ not only in the underlying assumptions but also in the predictions they make. For static noncooperative games, the most prominent solution is the Nash equilibrium (Nash 1951); there are many extensions of this concept (summarized in van Damme 1987). Other approaches to the solution of noncooperative games are those of Harsanyi and Selten (1988) and Fraser and Hipel (1984).

For static cooperative games, there are several classes of solution concepts. One prominent class consists of solutions that predict outcomes which are collectively rational (i.e., imputations). Included in this class are the core (Aumann 1961; Gillies 1959), the Shapley value (Shapley 1953), and the nucleolus (Schmeidler 1969). Other solutions in this class are the disruption nucleolus (Gately 1974; Littlechild and Vaidya 1976), the disruption value (Charnes et al. 1978), the p-center solution (Spinetto 1974), and the aspiration solution (Bennett 1983). Another class of solutions for cooperative games includes concepts that make payoff predictions contingent upon the coalition structures that form during play; these payoff allocations are usually coalitionally rational. Included are the M1(i) bargaining set (Aumann and Dreze 1974; Aumann and Maschler 1964), the competitive bargaining set (Horowitz 1973), the kernel (Davis and Maschler 1965), the Myerson–Shapley solution (Aumann and Myerson 1988; Myerson 1977), the equal division kernel (Crott and Albers 1981), and the alpha-power solution (Rapoport and Kahan 1982). Recently, a third class of solutions has emerged for cooperative games. Solutions in this class attempt not only to determine endogenously which coalition structure(s) will emerge but also to specify the associated payoffs to players. One solution in this class is the central-union theory (Michener and Au 1994; Michener and Myers 1998), which predicts coalition formation probabilistically. Another solution in this class is the viable proposals theory (Sengupta and Sengupta 1994).

## EXPERIMENTAL STUDIES OF GAMES

Laboratory experimentation on two-person and *n*-person games commenced in the early 1950s (e.g., Flood 1952) and it continues to the present. Some gaming studies are primarily descriptive in nature, whereas others investigate the predictive accuracy of various solution concepts.

**Experiments of Two-Person Games.** Investigators have conducted literally thousands of experiments on two-person games. Most of these treat noncooperative games, although some do treat cooperative games in various forms. Some studies investigate constant-sum games, whereas others treat non-constant-sum games (primarily such archetypal games as the prisoner's dilemma, chicken, battle of the sexes, etc.).

The major dependent variables in the two-person studies are the strategies used by players (particularly the frequency of cooperative choices) and the payoffs received by players. Independent variables include the type of game, strategy of the confederate, information set, interpersonal attitudes of players, sex of players, motivational orientation of players, and magnitude and form of payoffs.

Some of this research seeks to understand how differences in game matrices affect play (Harris 1972; Rapoport et al. 1976). Another portion describes how players' strategies vary as a function of the confederate's strategy (i.e., partner's history of play over time); this is reviewed in Oskamp (1971). Another portion of this work investigates the extent to which predictions from the minimax theorem approximate observed payoffs in constant-sum games; Colman (1982, ch. 5) reviewed these findings. Still other work covers cooperative bargaining models; Roth (1995) reviewed research on bargaining experiments. Some experimentation on two-person games has addressed the impact of players' value orientation on cooperation (McClintock and Liebrand 1988; Van Lange and Liebrand 1991). General reviews of experimental research on two-person games appear in Colman (1982), Komorita and Parks (1995), and Pruitt and Kimmel (1977).

**Experiments of** **n** **-Person Noncooperative Games.** There are several lines of experimentation on *n*-person noncooperative games. One line investigates multiperson compound games derived from 2 × 2 matrices (e.g., *n*-person chicken, *n*-person battle of the sexes, etc.). Important among these is the *n*-person dilemma (NPD) game, wherein individually rational strategies produce outcomes that are not collectively rational. The NPD serves as an abstract model of many phenomena, including conservation of scarce natural resources, voluntary wage restraint, and situations involving the tragedy of the commons (Hardin 1968; Hartwick and Yeung 1997; Moulin and Watts 1997). The literature contains many experimental studies of the NPD and other social dilemmas (e.g., Liebrand et al. 1992; Rapoport 1988). In addition to varying the payoff matrix itself, studies of this type investigate the effects of such factors as group identity, self-efficacy, perceptions of other players, value orientation, uncertainty, and players' expectations of cooperation. Reviews of research on NPDs and similar games appear in Dawes (1980), Kollock (1998), Komorita and Parks (1999), Liebrand and colleagues (1992), Liebrand and Messick (1996), and Messick and Brewer (1983).

A related line of research is that by experimental economists on markets and auctions (Smith 1982). This work investigates market structures (such as competitive exchange, oligopoly, and auction bidding) in laboratory settings (Friedman and Hoggatt 1980; Plott and Sunder 1982). Many of these structures can be viewed as noncooperative games. Plott (1982) provides a review of studies investigating equilibrium solutions of markets—the competitive equilibrium, the Cournot model, and the monopoly (joint maximization) model.

There is an increasingly large experimental literature on auctions, some of which is game-theoretic in character. Since auctions usually entail incomplete information (buyers have private information about their willingness to pay and ability to pay), these studies investigate the effects on bidding behavior of such variables as differential information, asymmetric beliefs, and risk aversion. They also investigate different institutional forms, such as English auctions, Dutch auctions, double auctions, and sealed bid–offer auctions (e.g., Cox et al. 1984; Smith et al. 1982). Reviews of the theoretical literature on auctions appear in Engelbrecht-Wiggans (1980), Laffont (1997), and McAfee and McMillan (1987). Kagel (1995) provides a survey of experimental research on auctions.

**Experiments of** **n** **-Person Weighted Majority Games.** Weighted majority games are an important subclass of cooperative, sidepayment, simple games. They serve as models of legislative or voting systems. Theorists have developed many special solution concepts for these games. Early theories applicable to weighted majority games are the minimum power theory and minimum resource theory (Gamson 1961). Riker's size principle predicts the formation of minimal winning coalitions in these games. Other theories for weighted majority games include the bargaining theory (Komorita and Chertkoff 1973; Kravitz 1986) and the equal excess model (Komorita 1979). The bargaining theory posits that players in a coalition will divide payoffs in a manner midway between equality and proportionality to resources (votes) contributed. The equal excess model is similar but uses the equal excess norm instead of proportionality.

Numerous experiments on coalition bargaining in weighted majority games have tested these and related theories (e.g., Cole et al. 1995; Komorita et al. 1989; Miller and Komorita 1986). Results of these studies generally support the bargaining theory and the equal excess model over the others, although all have deficiencies. Reviews of some experiments in this line appear in Komorita (1984) and Komorita and Kravitz (1983).

**Experiments of Other** **n** **-Person Cooperative Games.** Beyond NPD and weighted majority games, investigators have studied a wide variety of *n*-person cooperative games in other forms. The primary objective of the work is to discover which game-theoretic solution concepts predict most accurately the outcomes of these games.

Numerous studies have investigated cooperative sidepayment games in characteristic function form (e.g., Michener et al. 1986; Murnighan and Roth 1980; Rappaport 1990). Other studies have investigated similar games in strategic form. This work shows that in games with empty core, solution concepts such as the nucleolus and the kernel predict fairly well; in games with a nonempty core, however, the Shapley value is often more accurate. Reviews of parts of this research appear in Kahan and Rapoport (1984), Michener and Potter (1981), and Murnighan (1978).

Other studies have investigated cooperative nonsidepayment games. Some of this research pertains to bargaining models in sequential games of status (Friend et al. 1977). Other research tests various solution concepts (such as the core and the lambda transfer value) in nonsidepayment games in strategic form (McKelvey and Ordeshook 1982; Michener et al. 1985; Michener and Salzer 1989).

Another line of experimentation on cooperative nonsidepayment games is that conducted by political scientists interested in committee games or spatial voting games. These are *n*-person voting games in which policies are represented as positions in multidimensional space. For the most part, this research attempts to test predictions from alternative solution concepts (Ferejohn et al. 1980; Ordeshook and Winer 1980). Some of this work has led to new theories, such as the competitive solution (McKelvey and Ordeshook 1983; McKelvey et al. 1978), and to further developments regarding established ones, such as methods for computing the Copeland winner (Grofman et al. 1987). Experimental research on spatial games is reviewed in McKelvey and Ordeshook (1990).

## DYNAMIC GAMES

Although early developments in game theory centered primarily on static games (i.e., games in which interaction among players is single-period or single-play in nature), many subsequent developments have addressed *dynamic games* occurring over time. In a dynamic game, time (or stage) is an important consideration in strategy, and the choices and actions of players at any stage are conditional on the history of prior choices in the game. There is a growing theoretical literature on various classes of dynamic games, including repeated games, differential games, and evolutionary games. Introductions to the topic of dynamic games appear in Friedman (1990), Fudenberg and Tirole (1991), Owen (1982), and Thomas (1984). The empirical literature on dynamic games is still small relative to that on static games, although experimental studies of repeated games appear increasingly often.

The term *supergame* refers to a sequence of (ordinary) games played by a fixed set of players. One important type of supergame is the *repeated game*, wherein the same constituent game is played at each stage in the sequence. For instance, if some players play a prisoner's dilemma game again and again, they are engaging in a repeated game. At this point in historical time, the dominant paradigm for the study of dynamic strategic behavior is that of repeated games. Certain repeated games are of interest because they allow collectively rational outcomes to result from noncooperative equilibrium strategies. Axelrod (1984) has analyzed the development of cooperation in repeated games. Selten and Stoecker (1983) have used a learning theory approach to model end-game behavior of players in repeated prisoner's dilemma games. Aumann and Maschler (1995) have studied repeated games with incomplete information. A survey of literature on repeated games appears in Mertens and colleagues (1994a,b,c).

Theorists have developed various solution concepts applicable to repeated games and multistage games. Among these are the backward induction process, the subgame perfect equilibrium (Selten 1975), and the Pareto perfect equilibrium (Bernheim et al. 1987). Cronshaw (1997) describes computational techniques for finding all equilibria in infinitely repeated games with discounting and perfect monitoring.

Another class of dynamic games is the *differential game*, played in continuous time. Much of the literature on differential games focuses on the two-person zero-sum case. Some applications of differential games are military, such as pursuit games, where the goal of, say, a pursing aircraft is to minimize time or distance required to catch an evading aircraft (Hajek 1975). The classic works on differential games include Friedman (1971) and Isaacs (1965). Models of differential games with more than two players are discussed in Leitman (1974). Other useful works on differential games include Basar and Bernard (1989) and Lewin (1994). The vector-valued maximin for these games is discussed in Zhukovskiy and Salukvadze (1994).

Biologists and economists have used game theoretic concepts to study *evolutionary games*, which are dynamic models of social evolution that explain why certain inherited traits (i.e., behavioral patterns) arise in a human or animal population and remain stable over time. In some evolutionary games (especially those with animal populations), the individuals are modeled as having neither rationality, nor conscience, nor expectations, so strategy selection and equilibrium derive from behavioral phenotypes rather than from rational thought processes. Models of this type often incorporate such phenomena as mutation, acquisition (learning), and the consequences of random perturbations. Theorists have advanced various concepts of evolutionary stability and evolutionarily stable strategies (Amir and Berninghaus 1998; Bomze and Potscher 1989; Gardner et al. 1987; Maynard Smith 1982). Summaries and extensions of work on evolutionary games appear in Bomze (1996), Friedman (1991, 1998), Samuelson (1997), and Weibull (1995).

## INSTITUTIONAL ANALYSIS VIA GAME THEORY

Economists and political scientists have long used game theory in the analysis of social institutions. In work of this type, an analyst specifies an institution (such as a Cournot oligopoly or an approval voting system) in hypothetical or ideal-typical terms and then applies game-theoretic solution concepts to see which payoff allocation(s) may result at equilibrium. Through this approach, an analyst can compare the outcomes of alternative institutional forms with respect to stability, efficiency, and fairness. Broad discussions and reviews of this literature appear in Schotter (1981, 1994), Schotter and Schwodiauer (1980), and Shubik (1982, 1984).

**Economic Institutions.** Von Neumann and Morgenstern (1944) were among the first to explore the role of *n*-person game theory in economic analysis. Since that time, economists have analyzed a variety of institutions in game-theoretic terms, including *oligopoly and other imperfect markets*. Markets in which there are only a few sellers (oligopoly), two sellers (duopoly, a type of oligopoly), one buyer and one seller (bilateral monopoly), and so on, lend themselves to game-theoretic analysis because the payoffs to each player depend on the strategies of the other players.

Economists have modeled oligopolies both as noncooperative games and as cooperative games. Several analyses of oligopolies as noncooperative games show that the standard Chamberlin price-setting strategy is equivalent to a Nash equilibrium in pure strategies (Telser 1972). Beyond that, various analyses have treated oligopoly as a noncooperative multistage game. These analyses have produced generalizations concerning the effects of adjustment speed, demand and cost functions, and incomplete information regarding demand on the dynamic stability of the traditional Cournot solution and other equilibria (Friedman 1977; Radner 1980).

Analyses of collusion among oligopolists usually view this as a cooperative game. These treatments analyze outcomes via such solution concepts as the core or the bargaining set (Kaneko 1978). Reviews and discussions of game-theoretic models of oligopoly appear in deFraja and Delbono (1990), Friedman (1983), Kurz (1985), and Shubik (1984). Game theoretic literature on collusive equilibria is surveyed in Rees (1993).

A second topic of interest to game-theoretic economists is general equilibrium in a *multilateral exchange economy*. An early paper (Arrow and Debreu 1954) modeled a general competitive exchange economy (involving production, exchange, and consumption) as a noncooperative game, and then showed that a generalized Nash equilibrium existed for the model. Other theorists have modeled a multilateral exchange economy as a cooperative *n*-person game. Shubik (1959) showed that the core of such a game is identical to Edgeworth's traditional contract curve. More generally, Debreu and Scarf (1963) showed that the core converges to the Walrasian competitive equilibrium. Telser (1996) discussed the core as a tool for finding market-clearing prices in economies with many inputs and outputs. Roth and Sotomayor (1990) discussed the matching problem in labor and marriage markets.

By representing an economy as an *n*-person cooperative game, analysts can investigate general equilibrium even in markets that do not fulfill the neoclassical regularity assumptions (e.g., convexity of preferences, divisibility of commodities, absence of externalities) (Rosenthal 1971; Telser 1972). Beyond that, some economists model general equilibrium in game-theoretic terms because they can introduce and analyze a variety of alternative institutional arrangements. This includes, for instance, models with or without trade, with or without production, with various types of money (commodity money, fiat money, bank money, accounting money), and with various types of financial institutions (shares, central banks, bankruptcy rules, and the like) (Dagan et al. 1997; Dubey and Shubik 1977; Karatzas et al. 1997). Depending on the models used, such cooperative solutions as the core, Shapley value, and nucleolus play an important role in these analyses, as does the Nash noncooperative solution.

A third concern of game-theoretically oriented economists is the analysis of *public goods and services* (e.g., bridges, roads, dams, harbors, libraries, police services, public health services, and the like). Of special relevance is the pricing and cross-subsidization of public utilities; a central issue is how different classes of customers should divide the costs of providing public utilities.

Theorists can model such problems by letting the cost function of a public utility determine the characteristic function of a (cooperative) cost-sharing game. Games of this type are amenable to analysis via such solution concepts as the Shapley value (Loehman and Whinston 1974), the nucleolus (Littlechild and Vaidya 1976; Nakayama and Suzuki 1977), and the core (Littlechild and Thompson 1977), each of which represents alternative cost-sharing criteria. Moulin (1996) compared alternative cost-sharing mechanisms under increasing returns. Other work has used game-theoretic concepts to model negotiation with respect to provision of public goods (Dearden 1998; Schofield 1984a). A survey of experimental research on public goods appears in Ledyard (1995).

**Political Institutions.** Like economists, political scientists have analyzed a variety of institutions in game-theoretic terms. One broad line of work—termed the study of *social choice*—investigates various methods for aggregating individual preferences into collective decisions (Moulin 1994). Of special concern is the stability of outcomes produced by alternative *voting systems* (Hinich and Munger 1997; Nurmi 1987; Ordeshook 1986). Analysts have studied many different voting systems (e.g., majority voting, plurality voting, weighted voting, approval voting, and so on.) Strikingly, this work has demonstrated that in majority voting systems where voters choose among more than two alternatives, the conditions for equilibria (i.e., the conditions that assure a decisive winner) are so restrictive as to render equilibria virtually nonexistent (Fishburn 1973; Riker 1982).

Topics in voting include the detailed analysis of cyclic majority phenomena (generalized Condorcet paradox situations), the analysis of equilibria in novel voting systems such as weighted voting (Banzhaf 1965) and approval voting (i.e., a method of voting wherein voters may endorse as many candidates as they like in multicandidate elections) (Brams and Fishburn 1983), and the development of predictive solution concepts for voting systems that possess no stable equilibrium (Ferejohn and Grether 1982).

A second line of work by game theorists concerns the strategic manipulation of political institutions to gain favorable outcomes. One topic here is the consequences of manipulative *agenda control* in committees (Banks 1990; Plott and Levine 1978). Another topic is the effects of *strategic voting*—that is, voting in which players strive to manipulate the decision by voting for candidates or motions other than their real preferences (Cox and Shugart 1996; Feddersen and Pesendorfer 1998; Niemi and Frank 1982). Analyses by Gibbard (1973) and Satterthwaite (1975) showed that no voting procedure can be completely strategy-proof (in the sense of offering voters no incentive to vote strategically) without violating some more fundamental condition of democratic acceptability. In particular, any voting mechanism that is strategy-proof is also necessarily dictatorial. An important issue here is how to design voting systems with at least some desirable properties that encourage sincere revelation of preferences.

A third broad line of work concerns the *indexing of players' power* in political systems (Owen 1982, ch. 10). Frequently this entails assessing differences in a priori voting strength of members of committees or legislatures. For example, a classic study applied the Shapley–Shubik index of power to the U.S. legislative system and assessed the relative power of the president, senators, and representatives (Shapley and Shubik 1954). Other measures of power include the Banzhaf–Coleman index (Banzhaf 1965; Dubey and Shapley 1979) and Straffin's probabilistic indices (Straffin 1978).

A fourth game-theoretic topic is *cabinet coalition formation*, especially in the context of European governments. This topic is of interest because political fragmentation can produce instability of cabinet coalitions, which in turn can lead to collapse of entire governments. Work on this problem ties in with that on spatial voting games and weighted majority games, discussed above. Some cabinet coalition models stress policy (or ideological) alignment among members, while other models stress the transfer of value (payoffs) among members. Important theoretical models include DeSwaan (1973), Grofman (1982), Laver and Shepsle (1996), and Schofield (1984b).

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H. Andrew Michener