Games

Abstract

This entry begins by setting games within the general framework to demonstrate their conceptual importance for describing this activity of humankind. Then the concept of merging economics and mathematics into what is called game theory is developed. Descriptions of important properties, definitions, and typical examples of games are provided.

Definition

A game is a structured interaction of at least two players. It consists of rules that set the institutional framework of possible moves. These lead to outcomes that are subject to individual preferences, which generate the incentives to play. If individual ends are incompatible, dilemmata develop, which make the underlying strategic interaction particularly visible.

Games and Science

Even before John von Neumann and Oskar Morgenstern (1944) merged mathematical and economic theory in their groundbreaking work Theory of Games and Economic Behavior games were attracting interest in other fields. For instance, Johan Huizinga (1933) looked at games from a psychological perspective showing that they are a fundamental force and a formative element of culture as defined by philosophical anthropology. Games date back to a time before the development of humankind: Many creatures, especially primates, but also other mammals as well, show game-related behavioral patterns as an expression of rivalry. From a phylogenetic perspective, animals play, and from an ontogenetic perspective, games are incorporated into the existence and the development of the personal individuality of humankind. Games represent a field of tension between conflict – mankind’s aggression – and cooperation, the attempt to bind interests. If we say “a game is captivating,” then we metaphorically combine these antagonistic aspects in one sentence. In history and social sciences, managing conflict and mapping it in parlor games have always accompanied human development. In fact, John von Neumann (1928) looked at this issue in his first book on game theory. The greatest works on strategy, which are often compared today to game-theoretic models, were developed by scholars like Sun Zi (544–466 BC) from China or Carl von Clausewitz (1780–1831) from Germany. The Art of War (~500 BC, 2003, 2007) and On War (Vom Kriege, 1832) are two classic books about leadership under conditions that involve strategic, operational, or tactical interaction – an aspect of humankind that is also found in important religious works such as the Talmud, the Bible, and the Koran.

The work of John von Neumann (1928) and Oskar Morgenstern triggered not only a broad scientific development of game theory, such as the work by John Nash (1950) on equilibria published half a decade later, it also triggered a very strict observation of reality and its dilemma situations. On the level of political thinking, the concept of zero-sum games is perhaps the best-known outflow and potentially the least understood. In terms of the military, the concept of a credible threat during the Cold War was the direct result of game-theoretic thinking.

Economic and Political Aspects of Game Theory

If human rivalry is to be oriented toward a common good, and if the social contract written by societies is to be productive, the distinction between institutional rules and the moves allowed within this framework is of utmost importance. This relates to extending competition in markets to the idea of competition among institutions, especially public institutions. According to the philosophy of Immanuel Kant (1787) and, in particular, his concept of the “categorical imperative,” economic institutions should be designed in such a way that the maxim of their implementation can be accepted as a general rule or regulation. Thus, dilemmata reach beyond individual rivalry, extending to the interaction between institution setters, institutions, and their efficiency in relation to other institutions, thus binding together economics and law. This lies at the core of the very successful German social market economy in the tradition of Walter Eucken (1952) and Ludwig Erhard (1957). Rules are to be set so that they are self-enforcing and follow mankind’s open incentive concept. In such a world, games have two properties: rivalry over better arrangements at the higher institutional level and rivalry within the economic or social world below this roof. This addresses two aspects accentuated by game theory: hierarchy and dynamics. Who and how are the rules set for games, how open are they to long-term change, and who is in charge? Furthermore, with regard to moves (the complete set of which is called strategies), how can players learn to distinguish between “good” moves and “bad” moves? The final chapter will take up this subject using institutional competition as an example.

The rest of the entry is structured as follows: First we will look at the information side of game theory and the prerequisites for efficient activities. Then we will look at how one type of game may switch to other type when circumstances change. Finally, we will look at dynamic contexts.

The General Structure of Games

A game is a model of strategic (or tactical) interaction between players who have sets of possible actions. Since the model captures this interaction between the players, outcomes are interdependent. Dixit and Nalebuff (2010) is an introductory text that requires little formal experience. Standard textbooks are Rasmusen (2006) or Osborne and Rubinstein (1994). Binmore (1991) adds humor to theory. Players come up with strategies s and have preferences that are modeled by ordinal payoffs or cardinal functions, for instance, production functions p(.) or utility functions u(.). In its extensive form, the game starts with two possible moves from the first player’s strategy set (s₁¹ and s₁²). This is followed by the moves of the second player. Figure 1 illustrates the sequential structure.

Games, Fig. 1

Extensive form of a two-player game

One of the best-known games is the prisoner’s dilemma. The fundamental idea is straightforward: A murder has been committed and the police apprehend two suspects tramping in the area. They lock them up separately and have them interrogated by an attorney. The vagrants would be best off if they both remained silent; then they would be sentenced to only 1 year each for vagrancy. However, if one of them finks (cheats, defects) and accuses the other, he will be set free and the other will be jailed for 9 years. If both accuse the other person, they will both be jailed for 6 years. To all extent and purposes, it would be socially efficient to remain silent. However, under the absence of coordination, the best possible strategy is to accuse the other, which, however, produces an inferior outcome (Fig. 2).

Games, Fig. 2

Prisoner’s dilemma game

If both defendants remain silent, the outcome is better than if they accuse each other (both payoffs are strictly higher values). The best strategies for one player against the other are underlined. However, this advantage does not exist with respect to the other two pairs of payoffs. In addition, the payoff resulting from mutual accusation remains unchanged in the sense that neither prisoner can deviate without disadvantaging himself. This leads to the following definition:

A set of player strategies is a Nash equilibrium if no deviation of any individual player’s strategies (i.e. given the strategies of the other players) results in a better outcome for that player.

Results in field D are thus underlined twice as they are the Nash equilibrium.

One of the most interesting games in the genre of prisoner’s dilemmas is the tragedy of the commons (Hardin 1968). In this game, a joint resource is distributed. An overexploitation and destruction of the resource can only be prevented if all parties agree to the terms of exploitation. However, under the absence of coordination, each individual will seek to achieve his/her own optimal outcome, and the resource will break down. The resource-efficient outcome will only be achieved if there is a high enough expectancy to cooperate. Figure 3 shows that this is the case if 60% act in consort. If player A then decides to defect, he will be worse off (6.6 instead of 6.8 points). This is not yet the case when only 30% act in consort.

Games, Fig. 3

Tragedy of the commons game

There are many interesting dilemma situations: The chicken game models a situation where two cars approach each other on a narrow road with the question being, which car gives way to which? The battle of the sexes models a situation where a couple wants to spend their weekend away from home; he wants to go to a soccer game, while she wants to go to the theater. Both strategies on their own are incompatible with one another, and there is no joint utility-maximizing outcome. The problem in this game is that mixed strategies are not possible. A compromise can only be found, and socially efficient results can only emerge, if a mix between the two is possible, for instance, if two weekends are taken into consideration.

From these discussions, two other notions can easily be derived:

• In zero-sum games, the extent of the loss on the one side is identical to the gain on the other side.

• Zero-sum games are a special variant of constant sum games.

• In policy discussions, people think situations are zero-sum situations even when they truly are not. For instance, following Ricardo’s classic theory of comparative costs, trade is a redistribution that makes both parties better off.

Games with Functions as Payoffs

Often, games are set within a dynamic context as they develop in an evolutionary way or are played repetitively. Players are then able to gain experience, which may encourage them not to fall into inefficient Nash equilibria, e.g., into the prisoner’s dilemma trap. Another aspect is the changing of payoffs in dynamic utility or production functions, as in the following case which combines the economics of competition with the legal side of good regulations and governance. The example follows (Blum et al. 2005, section 7.2.2). In a competitive environment, the antitrust organization sets penalties, p, if competition rules are broken. The standard reward from competition is s ≥ 0; it becomes zero if all players violate competition laws. An extra externality or fringe profit, f ≥ 0, emerges if all players comply with the rules. The effort of players is given as e ≥ 0 and the related disutility is g(e) ≥ 0 (Fig. 4).

Games, Fig. 4

Competition game

Table 1 lists three types of games and the structure of their payoffs, to which we have added a social-optimum outcome.

Games, Table 1 Basic game structures

If we evaluate the outcomes, we obtain:

• Prisoner’s dilemma: (player 1, B₁ > A₁ > D₁ > C₁; player 2, C₂ > A₂ > D₂ > B₂)

  $$ {\mathrm{B}}_1>{\mathrm{A}}_1\left(\mathrm{or}\ {\mathrm{C}}_2>{\mathrm{A}}_2\ \mathrm{resp}.\right)\iff s-p>s+f-g(e)\iff f<g(e)-p. $$

  (P1)
  $$ {\mathrm{A}}_1>{\mathrm{D}}_1\left(\mathrm{or}\ {\mathrm{A}}_2>{\mathrm{D}}_2\ \mathrm{resp}.\right)\iff s+f-g(e)>-p\iff f>g(e)-\left(p+s\right). $$

  (P2)
  $$ {\mathrm{D}}_1>{\mathrm{C}}_1\left(\mathrm{or}\ {\mathrm{D}}_2>{\mathrm{B}}_2\mathrm{resp}.\right)\iff -p>s-g(e)-p\iff g(e)>s. $$

  (P3)

• Chicken game: (player 1, B₁ > A₁ > C₁ > D₁; player 2, C₂ > A₂ > B₂ > D₂)

• $$ {\mathrm{B}}_1>{\mathrm{A}}_1\left(\mathrm{or}\ {\mathrm{C}}_2>{\mathrm{A}}_2\mathrm{resp}.\right)\iff s-p>s+f-g(e)\iff f<g(e)-p. $$

  (C1)

• $$ {\mathrm{A}}_1>{\mathrm{C}}_1\left(\mathrm{or}\ {\mathrm{A}}_2>{\mathrm{B}}_2\mathrm{resp}.\right)\iff s+f-g(e)>s-g(e)-p\iff f>-p. $$

  (C2)

• $$ {\mathrm{C}}_1>{\mathrm{D}}_1\left(\mathrm{or}\ {\mathrm{B}}_2>{\mathrm{D}}_2\mathrm{resp}.\right)\iff s-g(e)-p>-p\iff g(e)<s. $$

  (C3)

• Assurance game: (player 1, A₁ > D₁ > C₁ and A₁ > B₁; player 2, A₂ > D₂ > B₂ and A₂ > C₂)

• $$ {\mathrm{A}}_1>{\mathrm{D}}_1\left(\mathrm{or}\ {\mathrm{A}}_2>{\mathrm{D}}_2\mathrm{resp}.\right)\iff s+f-g(e)>-p\iff f>g(e)-\left(p+s\right). $$

  (A1)

• $$ {\mathrm{D}}_1>{\mathrm{C}}_1\left(\mathrm{or}\ {\mathrm{D}}_2>{\mathrm{B}}_2\mathrm{resp}.\right)\iff -p>s-g(e)-p\iff g(e)>s. $$

  (A2)

• $$ {\mathrm{A}}_1>{\mathrm{B}}_1\left(\mathrm{or}\ {\mathrm{A}}_2>{\mathrm{C}}_2\mathrm{resp}.\right)\iff s+f-g(e)>s-p\iff f>g(e)-p. $$

  (A3)

If we plot these results on a graph, we obtain the following result, which is illustrated in Fig. 5. It seems that good competition regimes depend on credible penalties; sufficient externalities for good competitive behavior that can be internalized, i.e., through long-term growth; and a limited disutility of effort, i.e., limited levels of transaction costs for compliance.

Games, Fig. 5

Efficient areas of the competition game. (Source: Blum et al. 2005, p. 191)

What Can Be Learned from Game Theory

Game theory is a formal tool that shows the extent to which the outcomes of different decisions interact, especially if players have opposing goals. It provides a rigorous analytical environment for evaluating efficiency in static or dynamic environments. Although the ability to model reality in a very concrete way is limited, the overall structural messages are extremely important in understanding how to manage dilemma situations. In addition, players should check whether the game is a true abstraction of reality – otherwise, failure can be bitter. In summer 2015, Greek finance minister Varoufakis made this experience when trying in vain to provoke the lender Troika in a poker-type manner and through impulsive behavior in order to improve his negotiation position, a strategy already proposed 50 years ago in the Strategy of Conflict by (Schelling 1960).

Cross-References

• Abuse of Dominance

• Blackmail

• Bounded Rationality

• Cartels and Collusion

• Coase and Property Rights

• Corruption

• Credibility

• Equilibrium Theory

• Governance

• Information Deficiencies in Contract Enforcement

• Institutional Economics

• Rationality