Chapter 4: Simultaneous Games

Overview

Interest will not lie.

17th-century proverb^[1]

Softening sales cause both ford and gm to reconsider their pricing.^[2] If both move at the same time, then they are playing a simultaneous-move game. Figure 11 presents an example of a simultaneous-move game. It’s important that you understand how to interpret games like this one, so please read this paragraph very carefully. In this game Player One chooses A or B, while at the same time Player Two chooses X or Y. Each player moves without knowing what the other person is going to do. The players’ combined moves determine their payoffs. For example, if Player One chooses A, and Player Two chooses X, then we are in the top left corner. Player One scores the first number, 10, as his payoff, and Player Two scores the second number, 5, as her payoff. If Player One chooses A, and Player Two chooses Y, then we would be in the top right box, and Player One scores 3 while Player Two scores 0. In all simultaneous-move games Player One will always be on the left, and Player Two will always be on top. The first number in the box will usually be Player One’s payoff and the second will be Player Two’s payoff. The players always know what score they will receive if they end up in any given box. The players, therefore, see Figure 11 before they move. Each player knows everything except what his opponent is going to do.

Figure 11

As with sequential games, in simultaneous games a player’s only goal is to maximize his payoff. The players are not trying to win by getting a higher score than their opponents. Consequently, Player Two would rather be in the top left box (where Player One gets 10, and Player Two gets 5) than the bottom right box (where Player One gets 1, and Player Two gets 4).

What should the players do in a simultaneous game? The best way to solve a simultaneous-move game is to look for a dominant strategy. A dominant strategy is one that you should play, regardless of what the other player does. In Figure 11, strategy A is dominant for Player One. If Player Two chooses X, then Player One gets 10 if he picks A, and 8 if he picks B. Thus, Player One would be better off playing A if he knows that Player Two will play X. Also, if Player Two plays Y, then Player One gets 3 if he plays A and 1 if he plays B. Consequently, Player One is also better off playing A if Player Two plays Y. Thus, regardless of what Player Two does, Player One gets a higher payoff playing A than B. Strategy A is therefore a dominant strategy and should be played by Player One no matter what.

A dominant strategy is a strategy that gives you a higher payoff than all of your other strategies, regardless of what your opponent does.

Player Two does not have a dominant strategy in this game. If Player Two believes that Player One will play A, then Player Two should play X. If, for some strange reason, Player Two believes that her opponent will play B, then she should play Y. Thus, while Player One should always play A no matter what, Player Two’s optimal strategy is determined by what she thinks Player One will do.

A dominant strategy is a powerful solution concept because you should play it even if you think your opponent is insane, is trying to help you, or is trying to destroy you. Playing a dominant strategy, by definition, maximizes your payoff.

To test your understanding of dominant strategies, consider this: Is stopping at a red light and going on a green light a dominant strategy when driving? Actually, no, it isn’t. You only want to go on green lights and stop on red lights if other drivers do the same. If you happened to drive through a town where everyone else went on red and stopped on green, you would be best off following their custom. In contrast, if everyone in this strange place were intent on electrocuting herself, you would be best served by not following the crowd. Avoiding electrocution is a dominant strategy; you should do it regardless of what other people do. In contrast, driving on the right side of the road is not a dominant strategy; you should do it only if other people also do it.

Let’s return to Ford and GM’s pricing game. Figures 12 and 13 present possible models for the auto pricing game. In these games Ford is Player One while GM is Player Two. In response to weakening sales, both firms can either offer a discount or not offer a discount. Please look at these two figures and determine how the firms’ optimal strategies differ in these two games.

Figure 12

Figure 13

In Figure 12, offering a discount is a dominant strategy for both firms since offering a discount always yields a greater profit. Perhaps in this game, consumers will purchase cars only if given discounts. Figure 13 lacks dominant strategies. If your opponent offers a discount, you are better off giving one too. If, however, your opponent doesn’t lower his prices, then neither should you. Perhaps in this game consumers are willing to forgo discounts only as long as no one offers them. Of course, if you can maintain the same sales, you are always better off not lowering prices. This doesn’t mean that neither firm in Figure 13 should offer a discount. Not offering a discount is not a dominant strategy. Rather, each firm must try to guess its opponent’s strategy before formulating its own move.

The opposite of a dominant strategy is a strictly stupid strategy.^[3] A strictly stupid strategy always gives you a lower payoff than some other strategy, regardless of what your opponent does. In Figure 12, not offering a discount is a strictly stupid strategy for both firms, since it always results in their getting zero profits. In a game where you have only two strategies, if one is dominant, then the other must be strictly stupid.

A strictly stupid strategy is a strategy that gives you a lower payoff than at least one of your other strategies, regardless of what your opponent does.

Knowing that your opponent will never play a strictly stupid strategy can help you formulate your optimal move. Consider the game in Figure 14 in which two competitors each pick what price they should charge. Player Two can choose to charge either a high, medium, or low price, while for some reason Player One can charge only a high or low price. As you should be able to see from Figure 14, if Player One knows that Player Two will choose high or medium prices, than Player One will be better off with high prices. If, however, Player Two goes with low prices, then Player One would also want low prices. The following chart shows Player One’s optimal move for all three strategies that Player Two could employ:

Figure 14

Table 1

Player Two’s Strategy	Player One’s Best Strategy*
High	High
Medium	High
Low	Low

*If he knows what Player Two is going to do.

When Player One moves, he doesn’t know how Player Two will move. Player One, however, could try to figure out what Player Two will do. Indeed, to solve most simultaneous games, a player must make some guess as to what strategies the other players will employ. In this game, at least, it’s easy to figure out what Player Two won’t do because Player Two always gets a payoff of zero if she plays low. (Remember, the second number in each box is Player Two’s payoff.) Playing high or medium always gives Player Two a positive payoff. Consequently, for Player Two, low is a strictly stupid strategy and should never be played. Once Player One knows that Player Two will never play low, Player One should play high. When Player Two realizes that Player One will play high, she will also play high since Player Two gets a payoff of 7 if both play high and gets a payoff of only 5 if she plays medium while Player One plays high.

Player Two will play high because Player One also will play high. Player One, however, only plays high because Player One believes that Player Two will not play low. Player Two’s strategy is thus determined by what she thinks Player One thinks that Player Two will do. Before you can move in game theory land, you must often predict what other people guess you will do.

^[1]Browning (1989), 389.

^[2]Wall Street Journal (July 2, 2002).

^[3]Game theorists use the phrase “dominated strategies,” but this phrase can be confusing since it looks and sounds like “dominant strategies.” Consequently I have decided to substitute the more scientific sounding “strictly stupid” for the term “dominated.”