Scissors and Rock

Thinking strategically is the focus of this chapter. The chapter includes references to Cesare Borgia, Machiavelli, Adam Smith, Napoleon, and Torstein Veblen. A Prisoners’ Dilemma situation and a Chicken Game are discussed to illustrate the essence of a game—with respect to understanding the functioning of markets. The example of Microsoft’s Internet Explorer challenging Netscape’s Navigator, by now historical, serves to demonstrate that in a strategic decision situation an agent cannot choose an outcome, independent of what the other agents do—in fact, a player chooses a strategy and not an outcome. Managers have to know game theory if they want to apply it to outsmart the competitor in strategic decision situations. An excursion into the world of neuroscience and mirror neurons concludes the chapter. Mirror neurons are of interest to strategic thinking as they serve as a mechanism of imitation in our brain and as a source of empathy.

A short introduction into the history of game theory shows game theory’s close relationship to mathematics and its roots in “games of conflict and war.” The basic concepts of game theory are discussed making use of the fundamental operations of summing up, subtracting, multiplying, and dividing. It is suggested that readers become familiar with concepts such as sets, functions, and vectors. These concepts help to make the message of the text more concise and easier to structure—and thus easier to access. A brief reference to probabilities concludes the chapter by referring to Émile Borel’sinfinite monkey theorem.

The Prisoners’ Dilemma serves to introduce and discuss the concepts of players, strategies, payoffs, game form (event matrix), game matrix, and game tree, all of which specify the normal form (or strategic form) and the sequential form of strategic interaction. The Prisoners’ Dilemma game is illustrated by real-world cases when a state (or crown) witness chooses the dominant strategy. In 2002, launched by the US Department of Justice to look into alleged price-fixing in the DRAM computer chip market, Micron executives received amnesty because they agreed to testify as state witnesses against the price cartel. The investigation resulted in fines to Samsung, Infineon, Hynix and Elpida Memory equaling altogether $731 million for illegal price-fixing, and thereby violating antitrust law. In April 2002, Sotheby’s chairman, Alfred Taubman, and its chief executive, Diana Brooks, were found guilty of conspiring with Christie’s to fix commissions. Mr. Taubman served ten months of a one-year prison sentence; Ms. Brooks was given six months’ house arrest, a $350,000 fine, and 1000 h of community service. She pleaded guilty to price-fixing and then testified against her boss, Alfred Taubman. No one was charged at Christie’s, which had blown the whistle on the commission-fixing. These results propose one to ask: “Who are the players,” and “Do they have dominant strategies.” With respect to the second question, we examined Sun Tzu’s strategies of war and Tosca’s tragic fate.

Here a definition of the Nash equilibrium is presented, regarded as the most prominent solution concept for non-cooperative games. Information requirements such as common knowledge of rationality (CKR) and consistent-aligned beliefs (CAB) are discussed. A historical note on Nash and his equilibrium concept accompanies the definition, including a reference to the movie “Beautiful Mind.” For illustration, the concept is applied to the QWERTY-DSK game of standardization—and to the Kama Sutra.

The chapter discusses the extensive form of a game when sequences of moves may matter. In general, the game tree is an adequate representation of the sequential structure of a game. Using the game tree representation, the implications of missing recall, solutions to sharing a cake, and a sequential form of the Battle of Sexes are analysed. Moves can also be ingredients of thought experiments and backward induction can stabilize strategy choices which do not constitute a Nash equilibrium. This is the message of the Theory of Moves discussed in the concluding section of the chapter.

If there are more than one Nash equilibria, then often players face a serious coordination problem: “too many and too few.” In the Market Congestion Game, an (equilibrium) outcome can be expected featuring a Market B, without sellers, while too many sellers crowd Market A, resulting in an inefficient outcome. In the Volunteer’s Dilemma, coordination is likely to fail because players want to profit off “somebody else” volunteering. The dilemma implies that you only benefit from the volunteering of others if you do not volunteer yourself.

Many of the world’s most outstanding theater plays derive their dramatic effect from the fact that the hero does not follow the path of action that corresponds to what the audience considers the dominant strategy. This holds for Brutus in Shakespeare’sJulius Cesar as well as for Schiller’s Wallenstein, both characters who hesitate to grab power and thereby make use of the possibility to escape their fate. However, if players do not have dominant strategies and there are more than one Nash equilibria so that players control alternative equilibrium strategies, then the decision problem becomes even more challenging. Selten’s trembling hand perfectness can be applied to select equilibrium strategies which are still adequate even when it is assumed that the other players deviate from their equilibrium strategies in the form of small ε trembles. Often this reduces the set of Nash equilibria and in some cases a singleton is left. In contrast to this operation, the concept of rationalizable strategies leads to an expansion of the set of justifiable strategy choices. A strategy x is rationalizable if it is a best reply to strategy y and y is a best reply to a strategy z. In a Nash equilibrium (x, y), x and y are mutually best replies. It is immediately understood that Nash equilibrium strategies are rationalizable, but not all rationalizable strategies are Nash equilibrium strategies.

In a sequential game, subgame perfectness selects equilibria such that players choose mutually best replies not only at the beginning of the game but also in every subgame. Consequentially, neither player has an incentive to deviate from the chosen equilibrium strategy in the course of the game. The sequence of moves of a strategy from the beginning until the final nodes is structured in a hierarchy of subgames. A true subgame starts with an information set containing one node only, i.e., there is perfect information. A market entry game is analyzed with a competitor deciding on whether to enter a monopoly market with the possibility that the monopolist will “fight,” e.g., lower the price such that both agents suffer from losses. Further, the equilibria of the Ultimatum Game and the Stag Hunt Game are discussed—also with respect to social trust. In this context, a model is presented to illustrate how reciprocity works.

The focus of this chapter is on the Iterated Prisoners’ Dilemma and Robert Axelrod’s “Tournament of Strategies.” If a Prisoners’ Dilemma is repeated and the repetitions are with “unforeseeable end,” then an equilibrium exists that implies “cooperation” in every round of the supergame. This result is the most prominent implication of the Folk Theorem formalizing “what we have always known.” An obvious condition of achieving this “favorable result” is that the players appreciate future benefits and do not discount them too heavily. Another condition is that players know that this also applies to their opponents. In Axelrod’s tournament, strategies are randomly matched in pairs. The strategies prescribe the decision in each period for a finite number of periods. For instance, the winning strategy, i.e., TIT-FOR TAT, proposes that the player “cooperates” in the first period and repeats this choice in the following periods as long as the strategy of the opponent chooses “cooperate.” However, if the strategy selects “defect,” then TIT-FOR TAT presents “defect” as well. The strategy resulting in the largest sum of payoff points wins. This condition is different than what defines success in the Iterated Prisoners’ Dilemma.

The analysis ofmixed-strategyNash equilibrium, offered in this chapter, builds on the concept of expected utility—which has been challenged by Allais, Kahneman-Tversky, and numerous experimental results. Here, it is applied as a theoretical tool to discuss the peculiarities of the Nash equilibrium when it is mixed as, e.g., in the Inspection Game. It is demonstrated that in the Nash equilibrium the strategy of player 1 is exclusively determined by the payoffs of player 2, and vice versa. Moreover, if the Maximin Solution of this game is in mixed strategies as well, then its payoffs are identical to the payoffs of the Nash equilibrium. In general, however, the prescribed strategies are different because in the Maximin Solution the strategy of player i depends exclusively on the payoffs of i. Arthur Miller’s “The Crucible,” the Stag Hunt Game, and Peter Handke’s “The Goalie’s Anxiety at the Penalty Kick” illustrate the results of this chapter.

If there are more than two players in a game, then there is a potential for forming coalitions. The Core is the most prominent solution concept in this case. It assigns payoff vectors to the players. However, the Core of an n-person game can be empty or contain an infinite number of possible outcome vectors. In this chapter, applying the Core to a network game illustrates this problem. As a consequence, alternative solution concepts are briefly discussed: the Stable Set concept, proposed by von Neumannand Morgenstern, as the Solution; Bargaining Sets that are based on objections and (successful) counterobjections; the Kernel and the Nucleolus. Results are confronted with the problem of competition and cooperation in the triad.

In this chapter, after specifying the bargaining problem à la Nash (1950), the Rubinstein game and its subgame-perfect Nash equilibrium are presented. The equilibrium is determined (1) by the shrinking of the cake to be distributed among the two players and (2) by how the shrinking is evaluated by the players. If players can make binding agreements, then the self-enforcing power of the equilibrium is no longer needed. The players can jointly decide on feasible payoffs to serve as a bargaining outcome. The Nash solution is the most prominent concept that supports such a decision; it is considered to be fair and reasonable. The Kalai-Smorodinsky solution is an alternative concept briefly discussed in the chapter. The Rubinstein game and the Nash solution “meet” in the Nash program. More generally, the Nash program asks for a non-cooperative game, like the Rubinstein game, to produce outcomes as suggested by a cooperative game of the Nash solution type. Of course, a non-cooperative game can result in cooperation.

This chapter is dedicated to Goethe’s price game which can be interpreted as a second-price sealed-bidauction. Nobel Laureate William Vickrey has shown for this type of auction that bidding one’s willingness-to-pay price is an optimal strategy. We do not know whether Goethe had a similar result in mind. In fact, his sealed-bid price functioned like a take-it-or-leave-it option. Given the historical circumstances, it implied price-fixing, thereby avoiding explicit bargaining. A discussion of the Revenue Equivalence Theorem adds some theory to Goethe’s auction setting. The theorem says that, in principle, first-price sealed-bidauctions, Dutchauctions, second-price sealed-bidauctions, and Englishauctions pick the same winners and have identical prices. There are other types of auctions as well, and some have rather peculiar effects. All-payauctions are recommended for exploiting bidders—as in collecting money for a charity. If you want to win, do not participate. English auctions can have peculiar results, too. You might be interested in paying a high auction price for Gauguin’s “Mata Mua” if you already have a number of Gauguin paintings in your collection. On May 9, 1989, the “Mata Mua” was auctioned by Sotheby at 24.2 million dollars, then being the highest price ever bid for a Gauguin painting. You might also be willing and able to pay a high auction price if you represent a consortium of the Federal Republic of Germany, the States of Lower Saxony and Bavaria, the Prussian Cultural Heritage Foundation, and private donators. On December 6, 1983, the gospel book of Henry the Lion (1129–1195) was auctioned and sold for a price of £8,140,000, then equaling 32.5 million German marks. There was a single bidder present in the auction hall, i.e., the representative of the consortium, and a second person calling bids from outside—possibly the seller.