Classification of 2x2 games and strategic business behavior.

Wang, X. Henry ; Yang, Bill Z.

1. Introduction

In the last twenty years or so game theoretic approach has been gradually introduced into textbooks of microeconomics, including business and managerial economics, at undergraduate as well as graduate levels when discussing the theory of oligopoly and other topics on the behavior of multiple decision makers. Most, if not all, texts at the level of principles have covered simple 2x2 games. Popular one-shot games include the prisoner's dilemma, the battle-of-the-sexes, and chicken games. These games are mainly used as illustrative examples to interpret the concepts of normal-form representation of a game, tactical interdependence in decision making, dominant strategy, Nash equilibrium, cooperation, coordination, and so on. This paper is intended to show that each of those illustrative games belongs to an exclusive category of general symmetric and anti-diagonally symmetric 2x2 games (1), and to demonstrate how 2x2 games can also be used to explain strategic business behavior of firms in addition to the aspect of tactical interdependence in decision making. (2)

General symmetric 2x2 games are exhaustively classified into the game with dominant strategies, including the prisoner's dilemma and efficient dominant strategy games, the chicken game, and the coordination game (Eshel et al, 1998). And the well-known battle-of-the-sexes game gives an example of anti-diagonally symmetric 2x2 games. While cooperation and efficiency are the key issues in the prisoner's dilemma game, coordination and conflict of interest are the major concerns in the coordination game, the chicken game and the battle-of-the-sexes game. To move strategically may help a game player to solve these problems.

Strategic behavior is "a set of actions a firm takes to influence the market environment so as to increase its profit" (Carlton and Perloff, 1999, p. 332). Recently, "thinking strategically" (Dixit and Nalebuff, 1993) has become a part of the core themes in applied microeconomics courses for business as well as economics students. For a technical reason, however, this topic is normally skipped at lower levels. We show that 2x2 games can be used to explain a variety of business strategies such as "top dog," "puppy dog," "fat cat" and "lean-and-hungry look" (Fudenberg and Tirole, 1984). (3) Since 2x2 games have already been covered in the principles of microeconomics, this approach may help downgrade the "theory of business strategy" (Shapiro, 1989) that is currently taught only in specialized and more advanced courses such as industrial organization (Tirole, 1988, Carlton and Perloff, 1999, and Church and Ware, 1999, among others) and graduate microeconomic theory (Kreps, 1990, for example).

2. Symmetric 2x2 games: Classification and examples

In a general 2x2 game, there are two game players: player A and player B, who choose one strategy from the possibilities X and Y. Once both players have chosen their strategies, each player receives a corresponding payoff, determined by the combination of strategies the players have chosen. A general 2x2 game is usually described in a payoff matrix as shown in Figure 1.

For each pair of payoffs, the first one in a cell is to Player A, and the second one to Player B. For example, a(x, y) represents the payoff to player A when s/he chooses strategy X, and player B chooses strategy Y, etc. Both players prefer a larger payoff, and each player chooses a strategy solely on the basis of her/his own payoffs.

A 2x2 game is parameterized by eight payoffs whose relative values characterize the game. There are [81.sup.2] possibilities in terms of the relative positions of the payoffs of 2x2 games.

Proof: First, we show that for each player's four payoffs, there are 81 possible relative positions among them. Second, because there are two players, there are totally [81.sup.2] possible 2x2 games. So, the question becomes: Among a(x, x), a(x, y), a(y, x) a(y, y), how many possible relative positions are there?

Between a pair of real numbers X and Y, there are three possible cases: >, =, and <. Depending on whether there are any "=" among a(x, x), a(x, y), a(y, x) a(y, y), we have four possible cases:

(0) There is no equal sign. That is, all four numbers are different. There are 4! = 24 possibilities in this case.

(1) There is exactly one equal sign. It means that there is one pair plus two singles, or there are three distinct numbers. There are [C.sup.2.sub.4] = 6 ways to determine the pair. For each of such 6 cases, there are 3! = 6 relative positions for the pair and singles. So, there are 6*3! = 36 cases.

(2) There are exactly two equal signs in the array. In this case, there two sub cases: two pairs or a triplet plus a single. There are 6 ways to make two pairs and there are 4 ways to make a triplet plus a single. In either sub case, there are 2! positions since there are exactly two distinct values in the four entries. Consequently, there are totally (6+4)*2! = 20 positions.

(3) There are exactly three equal signs, i.e., a(x, x) = a(x, y) = a(y, x) = a(y, y). So there is only one possibility.

Therefore, there are totally 24 + 36 + 20 + 1 = 81 possibilities for Player A's payoffs. Similarly, there are 81 possibilities for Player B's payoffs. Put them together, there are totally 81x81 = [81.sup.2] possible 2x2 games in terms of relative values of payoffs. Q.E.D

This paper focuses on symmetric and anti-diagonally symmetric 2x2 games. In this section, we discuss the classification and the characteristics of symmetric 2x2 games. A 2x2 game given in Figure 1 is symmetric if a(x, x) = b(x, x), a(y, y) = b(y, y), a(y, x) = b(x, y), and b(y, x) = a(x, y). Without losing generality, it is assumed that a(x, x) > a(y, y) and a(y, x) > b(y, x). In terms of the relative values between a(x, x) and a(y, x), and between b(y, x) and b(y, y), Eshel et al (1998) classify symmetric 2x2 games into three types: (1) the game with dominant strategies, including the prisoner's dilemma game (4) and the efficient dominant strategy game, (2) the chicken game, (5) and (3) the coordination game, as summarized in Table 1. (6)

Now, we discuss each type of the games one by one as follows.

(1A) Prisoner's dilemma game. In this game, both players have a dominant strategy to choose Y that yields a larger payoff to each player regardless of the choice made by the other. In the dominant strategy equilibrium (Y, Y), however, both players end up with the payoff of a(y, y) = b(y, y), which is smaller than a(x, x) = b(x, x), the payoff they could earn if they both choose X instead. So, the key issue in the prisoner's dilemma game is that the dominant strategy prevents both from playing cooperatively and hence from achieving the efficient outcome. The following example illustrates this point.

Example 1. A pricing game. In a market for SUV there are two dealers competing in prices. They each can choose a high or a low price. This one-shot game has the payoff matrix shown in Figure 2 with the payoffs being their expected profits.

This is a prisoner's dilemma game: both dealers have a dominant strategy to set a low price, but they could be better off if both of them set high prices. At the level of principles of microeconomics, examples for applications of prisoner's dilemma game include R&D game, advertising game, quantity game, free-rider game, etc., in addition to the above pricing game. They all share a common point that each player has a dominant strategy and the equilibrium outcome is not Pareto efficient. And a key problem in such a game is how the two players can reach a win-win solution. The next example gives an answer to this question.

(1B) Efficient dominant-strategy game. In this game both players have a dominant strategy to choose X. And the dominant strategy equilibrium (X, X) is a Pareto efficient outcome. That is why this game so named. Let's see a numerical example for such a game.

Example 2. A pricing game with the meet-the-competition clause (MCC). Continued from Example 1, a meet-the-competition clause is introduced. Namely, when both dealers non-operatively advertise their prices, they also announce that they will meet the competitor's price if the latter is lower. Now, they each can choose to advertise a high price or a low price, anticipating that the final transaction price may differ if they have to lower it later. So, there are two prices for each dealer now: the announced price and the actual price charged in the transaction. Under the meet-the competition clause, the payoff matrix is shown in Figure 3. (7)

In this pricing game with MCC, both players have dominant strategies to announce, and thus actually sell at, a high price. By taking a seemingly aggressive pricing strategy like MCC, each dealer forces the other to set a high price--a Pareto efficient outcome. Hence, "putting in MCCs changes the game ... into one with a win-win element" (Brandenburger and Nalebuff, p. 171).

(2) Chicken game. In this game there are two asymmetric Nash equilibria (X, Y) and (Y, X). In a 2x2 game, a Nash equilibrium is a pair of strategies chosen by the two players, such that if either player chooses a strategy as in the pair, then the other player cannot get a larger payoff by choosing the strategy that is not in the pair. Between the two equilibria (X, Y) and (Y, X), the best outcome for each player is for it to choose Y while the other chooses X, whereas the worst outcome is for both players to choose Y, even worse than if both choose X. Let's cite the Boeing-Airbus game from Krugman (1987) to illustrate the feature of the chicken game.

Example 3. Boeing-Airbus game. Boeing and Airbus are each planning to produce a newly developed aircraft. The ultimate payoff to each firm also depends on what the other firm does. The payoff matrix is given in Figure 4 where the payoffs represent the expected profits of the firms.

In this example, there are two Nash equilibria: (Produce, Not Produce) and (Not Produce, Produce). Obviously, each firm prefers to be the monopolist to produce the new aircraft, but both would suffer heavily if they both produce it. A key issue in a chicken game is how a player can commit itself to producing so as to induce the other not to produce. Section 4 will discuss this problem.

(3) Coordination game. In this game, there are two Nash equilibria: (X, X) and (Y, Y). The major concern for the two players in a symmetric coordination game is coordination--the two players' choices match each other. Since a(x, x) = b(x, x) > a(y, y) = b(y, y), the two players have a common interest in reaching the more desired equilibrium, (X, X), rather than the inferior equilibrium (Y, Y). A business-related game is provided below to illustrate the point.

Example 4. Microsoft and Intel game. Microsoft produces operating system, Windows, and Intel makes computer chips, Pentium. At a stage of development they can choose to produce a "faster" model or "fast" model, and the "speed" of their products must match with each other. The payoff matrix is given in Figure 5 the payoffs representing the expected profits of the firms.

There are two Nash equilibria: (Faster, Faster) and (Fast, Fast). Obviously, they both prefer the equilibrium (Faster, Faster). This commonly preferred choice is usually called the 'focal point', even though the other equilibrium cannot be ruled out by the nature of Nash equilibrium.

3. Anti-diagonally symmetric 2x2 games

In this section, we discuss the classification of anti-diagonally symmetric games. In Figure 1, a 2x2 game is anti-diagonally symmetric if a(x, x) = b(y, y), a(y, y) = b(x, x), a(y, x) = b(y, x), and a(x, y) = b(x, y). Without losing generality, it is assumed that a(y, x) = b(y, x) [greater than or equal to] a(x, y) = b(x, y) and a(x, x) = b(y, y) [greater than or equal to] a(y, y) = b(x, x). Like symmetric games, we also have four possible types of 2x2 anti-diagonally symmetric games, as shown in the table below. But only two of them are really new and of interest: the battle-of-the-sexes game, and the pure anti-diagonally symmetric coordination game. We examine these two games in detail and illustrate them in examples in Table 2.

(4) The battle-of-the-sexes game. (8) There are two Nash equilibria in this game: (X, X) and (Y, Y). Like the symmetric coordination game, the major concern between the two players is coordination. However, in contrast to the symmetric coordination game in which the two players prefer the same equilibrium to the other, the two players have different preferences between the two equilibria, because a(x, x) = b(y, y) > a(y, y) = b(x, x). Thus, in this game, each player not only wants to avoid coordination failure, but also wants to play its favorable equilibrium. This latter point is similar to that in chicken game.

The following example is modified from Pindyck and Rubinfeld (2000, p. 481) to illustrate an application of the battle-of-the-sexes game in business.

Example 5. Car and engine producers' game. Race Car Motors is a major car producer and Far Out Engine is a major engine manufacturer. Small/big engines fit small/big car production, though they each can buy or sell engines from or to other market participants. However, even when their productions match each other, their profits are different. The payoff matrix is given in Figure 6 with the payoffs representing the expected profits of the firms.

In this game, there are two Nash equilibria: (Small, Small) and (Large, Large). While the major concern is for both products to match in their productions, Far Out Engine prefers small engines and Race Car Motors prefers big cars. This issue will be discussed in section 4.

(5) The pure asymmetric coordination game. There are two Nash equilibria: (X, Y) and (Y, X). Like the battle-of-the-sexes game, the major concern between the two players is possible coordination failure. In contrast to the battle-of-the-sexes game, however, the two players choose different actions in each of the two equilibria. Another key feature in this game that is different from that in the battle-of-the-sexes is the equilibrium payoffs for both players are the same in both equilibria. So, coordination is the only problem and there is no common or conflict interest between the two players. The following numerical example illustrates the game.

Example 6. Wait-or-redial game. When two friends, Yvonne and Zach, are talking to each other on the telephone, the line is off for some reason. They both want to continue to talk, though. It is an asymmetric coordination problem: one must redial and the other must wait. The payoff matrix can be given as shown in Figure 7 with the payoffs being interpreted as their utility.

In this game, there are two Nash equilibria: (Redial, Wait) and (Wait, Redial). The only problem is coordination: one must do something the other is not doing. No equilibrium is more desirable than the other. And there is no conflict of interest, either.

4. Strategic Business Behavior in 2x2 games

In the previous two sections we discussed the classifications of general symmetric and anti-diagonally symmetric 2x2 games. The structure of a game determines the features of the resulting equilibrium such as efficiency, cooperation, coordination failure and so on. In this section, we further address these issues and provide possible solutions to these problems by introducing strategic movement in 2x2 games. A variety of strategic business behaviors are illustrated in examples.

Fudenberg and Tirole (1984) and Tirole (1988, p. 325) classify business strategies into the following four types: (9)

Top dog--be big or strong to look tough or aggressive;

Puppy dog--be small or weak to look soft or inoffensive;

Fat cat--be big or strong to look soft or inoffensive;

Lean and hungry look--be small or weak to look tough or aggressive.

We now use examples in 2x2 games to explain all of the four business strategies.

Example 3 (cont.). Strategic trade policy (Krugman, 1987). A top-dog strategy. Recall that in Example 3--the Boeing and Airbus game, each player prefers to be the monopolist to produce the newly developed aircraft, but fears whether the other player will produce it as well. Hence, the key issue for each of them is how to commit itself to producing it so as to induce its rival not to produce it. Suppose that the European governments commit to subsidizing the Airbus as long as it produces, no matter how Boeing is going to do. Then the payoff matrix of the game is shown in Figure 8.

In this new game under subsidy, it is a dominant strategy for Airbus to produce and the resulting unique Nash equilibrium is (Not Produce, Produce), which favors Airbus, This example has the following characteristics:

1) The original game is a chicken game, in which there are two Nash equilibria. Each player prefers "hawk" to "dove," but worries if the other would also act like a hawk.

2) In order to become the only "hawk," a player can take a strategic movement to make an irrevocable commitment. With this movement, it then changes the game into a different one with a unique Nash equilibrium in which the strategic mover takes an aggressive posture while its rival plays softly. This is an example of the top-dog strategy.

Though the above two points are related to the top-dog strategy in a chicken game, the idea behind them is applicable to other games and strategies. We continue to give more examples of different business strategies in 2x2 games.

Example 7. Hotel game. (10) A puppy-dog strategy. An entrant is planning to build a new hotel near an existing hotel in a resort location. The new hotel owner can choose between large and small in size. If the new hotel is to be built large, then the price game after the entry between the incumbent and the entrant is given in Figure 9 with the payoffs being hotels' expected profits.

This is a prisoner's dilemma game--both hotels have a dominant strategy to set a low price, though they would be better off if both of them set high prices. A key issue for the new hotel owner is to avoid triggering a price war. If the new hotel chooses to build small as a commitment that it will not take too many customers away from the incumbent hotel even if it charges a low price, then the price game becomes shown in Figure 10.

When the new hotel is built small in capacity, the entrant is committed to a low price as the dominant strategy. But it does not trigger a price war since the incumbent may continue to enjoy a high price. As a result, the entrant can earn a higher profit than if it builds a large hotel. It is a strategic move that the entrant chooses to be small and inoffensive in the first stage so as not to trigger a price war in the second stage. This is an example of puppy-dog strategy.

Examples 1 and 2 (Cont.). Meet-the-competition clause (MCC). A fat-cat strategy. Recall that in Example 1--the car dealers' pricing game is a prisoner's dilemma game. When both firms introduce MCC, in Example 2, we see that the pricing game with MCC becomes an efficient dominant-strategy game. To announce to meet the competition seems to be a very aggressive posture. But it ends up with soft actions by both firms. Hence, the meet-the-competition clause is a competition-reducing business practice and it may help sellers reach tacit collusion. This is an example of fat-cat strategy. (11)

Another classic facilitating practice that softens competition is the most-favored-customer clause (MFCC). (12) It "guarantees a firm's current customers that they will be reimbursed the difference between the current price and the lowest price offered in the future (up to some specific date)" (Tirole, 1988, p. 330). Therefore, to cut price in the future will lower a firm's profit since it has to refund its previous customers who have paid a higher price. A numerical 2x2 game similar to the one for meet-the-competition clause can be constructed. And it gives another example of fat-cat effect.

Example 5 (Cont.). Car and engine producers' game. A lean-and-hungry look. Example 5 is a battle-of-the-sexes game. The car and the engine must match between the two firms, who have different preferences. Suppose that Far Out shuts down or destroys its big engine production capacity. Then it reduces its profit from big engine production and the payoff matrix becomes as shown in Figure 11

Now, Far Out Engine has a dominant strategy to produce small engines. It then forces Race Car to produce small cars and Far Out has improved its outcome of the game. Far Out plays strategically by seemingly putting itself at a disadvantage. This is a strategy of lean-and-hungry look.

In the above four examples, we have shown that all of the four business strategies in the taxonomy by Fudenberg and Tirole (1984) can be illustrated in simple 2x2 games. Basically, moving strategically changes a prisoner's dilemma game, a chicken game, or a battle-of-the-sexes game into another game with a unique Nash equilibrium that favors the strategic player.

5. Conclusions

The purpose of this note is two fold. First, we extend the classification of general symmetric 2x2 games (Eshel, et al, 1998) to include anti-diagonally symmetric 2x2 games. It allows us to also cover the broadly cited battle-of-the-sexes game and the pure coordination game as a special case of a general 2x2 game. The structure of each type of game and the feature of the equilibrium are examined. The potential problems concerning the equilibrium of those games such as efficiency, cooperation or coordination lead us to the second topic of the paper: to explain strategic business behavior in 2x2 games.

In current textbooks, 2x2 games are mainly employed to explain the aspect of tactical interdependence of decisions made by game players. We show that 2x2 games can also be used to explain strategic business behavior. Nowadays, how to think strategically in business decisions may have become one of the central themes that economics and business students need to learn. A challenging task in pedagogy for economic and business professors is how to teach it with the skills and tools available to students. Since 2x2 games have been covered in almost all principles microeconomics texts, this paper may provide a simple method to teach a variety of strategic business behaviors so as to downgrade the topic to the undergraduate levels without losing the essence.

FIGURE 1. A general 2x2 game

 Player B

 X Y

Player A X a(x, x), b(x, x) a(x, y), b(x, y)
 Y a(y, x), b(y, x) a(y, y), b(y, y)

FIGURE 2. Car dealers' pricing game

 Dealer 2

 High Price Low Price

Dealer 1 High Price 100, 100 10, 120
 Low Price 120, 10 70, 70

FIGURE 3. Car dealers' pricing game under the
MCC

 Dealer 2

 High Price Low Price

Dealer 1 High Price 100, 100 75, 70
 Low Price 70, 75 70, 70

FIGURE 4. A chicken game

 AIRBUS

 High Price Low Price

BOEING Not Produce 0, 0 0, 100
 Produce 100, 0 -10, -10

FIGURE 5. A coordination game

 Intel

 Faster Fast

Microsoft Faster 20, 20 0, 0
 Fast 0, 0 10, 10

FIGURE 6. A battle-of-the-sexes game

 Race Car Motors

 Small Cars Big Cars

Far Out Small Engines 60, 40 15, 15
Engines Big Engines 15, 15 10, 60

FIGURE 7. A pure asymmetric coordination game

 Zach

 Redial Wait

Yvonne Redial 0, 0 1, 1
 Wait 1, 1 0, 0

FIGURE 8. Strategic trade policy

 Intel

 Not Produce Produce

BOEING Not Produce 0, 0 0, 120
 Produce 100, 0 -10, 10

FIGURE 9. Hotel game: Price war

 New Hotel

 High Price Low Price

Incumbent High Price 8, 8 2, 10
Hotel Low Price 10, 2 4, 4

FIGURE 10. Hotel game: A puppy-dog strategy

 New Hotel

 High Price Low Price

Incumbent High Price 9, 5 8, 7
Hotel Low Price 10, 2 5, 3

FIGURE 11. A lean-and-hungry-look strategy

 Race Car Motors

 Small Cars Big Cars

Far Out Small Engines 60, 40 15, 15
Engines Big Engines 0, 10 0, 60

TABLE 1

Label Name of the game Relative values of parameters

(1A) Prisoner's dilemma: a(x, x) < a(y, x), b(y, x) < b(y, y)
(1B) Efficient dominant
 strategy a(x, x) > a(y, x), b(y, x) > b(y, y)
(2) Chicken a(x, x) < a(y, x), b(y, x) > b(y, y)
(3) Coordination game a(x, x) > a(y, x), b(y, x) < b(y, y)

TABLE 2

Label Name of the game Relative values of parameters

(4) The battle of the sexes: a(x, x) > a(y, x), b(x, x) > b(x, y)
(5) Pure coordination: a(x, x) < a(y, x), b(x, x) < b(x, y)
(6A) Asymmetric Prisoner's
 Dilemma: a(x, x) > a(y, x), b(x, x) < b(x, y)
(6B) Asymmetric efficient
 Dominant strategy: a(x, x) < a(y, x), b(x, x) > b(x, y)

Notes

(1.) By a symmetric matrix game, we mean its payoff matrix is symmetric around the diagonals, while by an anti-diagonally symmetric matrix game, we mean the payoff matrix is symmetric around the anti-diagonals. In a kxk payoff matrix, the anti-diagonals are those cells in ith row and jth column such that i + j = k + 1.

(2.) Though the term "strategic behavior" is employed in the chapter or the section for introductory game theory in most textbooks, it actually refers to the "interdependence" of tactical decisions in a one-shot game, rather than the strategic move that changes the environment in which game players continue to interact.

(3.) See section 4 for the definitions of those business strategies.

(4.) The prisoner's dilemma game is perhaps the first example of games that all principles texts may have employed. The standard story behind it is that two suspects in a crime are put into separate cells and they each can choose to confess or deny, which will determine how severe each would be penalized. The moral in this game is that there are gains from cooperation for both suspects to deny, but each of them has an incentive to confess no matter how the other does, which actually makes both of them worse off.

(5.) The chicken game is also called "hawk-dove" game by evolutionary game theorists and biologists. It is so named because in this game if one player behaves aggressively like a hawk, it is the best for the other acts softly like a dove, and vice versa. Though the "hawk" gets more than the "dove" does, a key feature in this game is that if both players behave like a hawk, they would end up being even worse than if they both act like a dove.

(6.) Clearly, this classification is exhaustive for symmetric 2x2 games.

(7.) Note that the price as a choice variable in the game means the announced price, which may be different from the price annually charged in the final transaction. A slightly higher profit (75 > 70) for a dealer when it announces a high price than when it announces a low price while its competitor sets a low price can be justified by the existence of uninformed consumers who may not know the other dealer offers a lower price.

(8.) This is another broadly cited example. The story behind it is about a couple who plan to go out for entertainment. While they have different preferences over the available programs, say, the husband likes ballet better while the wife prefers boxing, it is most important that they go out together. Hence, the key issue in this game is coordination between the two with interest conflict.

(9.) For an excellent survey and applications, see Tirole (1988, pp. 323-336).

(10.) The example is motivated from Tirole (1988, p. 329).

(11.) For an intuitive discussion, see Brandenburger and Nalebuff (1996, pp. 170-175).

(12.) For a seminal work on strategic analysis of MFCC, see Cooper (1986). See also Church and Ware (1999, p. 525), for example.

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X. Henry Wang * and Bill Z. Yang **

* Department of Economics, The University of Missouri-Columbia, Columbia, MO 65211

** Department of Finance and Economics, Georgia Southern University, Statesboro, GA 30460-8151 We thank an anonymous referee, and Hanfeng Chen for comments and suggestions on earlier versions of this paper.