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.' A Billionaire’s Political Strategy Warren Buffett once proposed using dominant and strictly stupid strategies to get both the Republicans and Democrats to support campaign finance reform. [4] He suggested that some eccentric billionaire (not himself) propose a campaign finance bill. The billionaire promises that if the bill doesn’t get enacted into law, then he will give $1 billion to whichever party did the most to support it. Figure 15 presents this game where each party can either support or not support the bill. Assume that the bill doesn’t pass unless both parties support it. The boxes show the outcome rather than each party’s score. Figure 15 In the game that Buffett proposed, supporting the bill is a dominant strategy, and not supporting it is strictly stupid. If the other party supports the bill then you have to as well or else they get $1 billion. Similarly, if they don’t support the reform, you should support the bill, and then use your billion dollars to crush them in the next election. Buffet’s plan would likely work, and not even cost the billionaire anything, because both parties would always play their dominant strategy. [4] Campaign for America (September 12, 2000). More Challenging Simultaneous Games Games involving dominant or strictly stupid strategies are usually easy to solve, so we will now consider more challenging games. Coordination Games How would you play the game in Figure 16? Obviously you should try to guess your opponent's move. If you're Player One you want to play A if your opponent plays X and play B if she plays Y. Fortunately, Player Two would be willing to work with you to achieve this goal so, for example, if she knows you are going to play A she will play X. In games like the one in Figure 16 the players benefit from cooperation. It would be silly for either player to hide her move or lie about what she planned on doing. In these types of games the players need to coordinate their actions. Figure 16 Traffic lights are a real-life coordination mechanism. Consider Figure 17. It illustrates a game that all drivers play. Two drivers approach each other at an intersection. Each driver can go or stop. While both drivers would prefer to not stop, if they both go they have a problem. Figure 17 Coordination games also manifest when you are arranging to meet someone, and you both obviously want to end up at the same location, or where you're trying to match your production schedule with a supplier's deliveries. Figure 18 shows a coordination game that two movie studios play, in which they each plan to release a big budget film over one of the next three weeks. Each studio would prefer not to release its film when its rival does. The obvious strategy for the studios to follow is for at least one of them to announce when its film will be released. The other can choose a different week to premiere its film, so that both can reap high sales. Figure 18 Technology companies play coordination games when they try to implement common standards. For example, several companies are currently attempting to adopt a high- capacity blue-laser-based replace-ment for DVD players. Consumers are more likely to buy a DVD replacement if there is one standard that will run most software, rather than if they must get separate machines for each movie format. Consequently, companies have incentives to work together to design and market one standard. In game theory land you do not trust someone because she is honorable or smiles sweetly when conversing with you. You trust someone only when it serves her interest to be honest. In traffic games a rational person would rarely try to fake someone out by pretending to stop at a red light only to quickly speed through the traffic signal. Since coordination leads to victory (avoiding accidents) in traffic games, you should trust your fellow drivers. In all coordination games your fellow player wants you to know her move and her benefits from keeping her promises about what moves she will make. The key to succeeding in coordination games is to be open, honest, and trusting. Trust Games Trust games are like coordination games except that you have a safe course to take if you're not sure whether your coordination efforts will succeed. Figure 19 illustrates a trust game. In this game, both players would be willing to play A if each knew that the other would play A as well. If, however, either player doubts that the other will play A, then he or she will play B. Playing B is the safe strategy because you get the same moderate payoff regardless of what your opponent does. Playing A is more risky; if your opponent also plays A, you do fairly well. Playing A when your opponent plays B, however, gives you an extremely low payoff. Figure 19 Figure 20 illustrates a trust game where both you and a coworker demand a raise. In this game your boss could and would be willing to fire one of you, but couldn't afford to lose you both. If you jointly demand a raise, you both get it. Alas, if only one tries to get more money, he gets terminated. Figure 20 In any trust game there is a safe course where you get a guaranteed payoff, and there is a risky strategy that gives you a high payoff if your fellow player does what he is supposed to. A simple example of a trust game is where two companies work together on a research project and both must complete their research for the project to succeed. The safe course for each company to take would be not to do any research, and thereby ensure that neither company has anything to lose in a joint venture. If your company does invest in the project, it receives a high payoff if the other firm fulfills its obligations. If, however, the other firm does not complete its research, then your firm loses its investment. Let's consider a trust game played by criminals. When two individuals hope to profit from an illegal insider-trading scheme, they enter a trust game. Imagine you work at a law firm. Through your job you learn that Acme Corporation intends to buy Beta Company. You and only a few others know of Acme's plans. When Acme's takeover plans become public in a few days, you're sure that the value of Beta stock will rise. Obviously, you would like to buy lots of Beta stock before its price increases, but unfortunately, if you did buy Beta stock you would be guilty of insider trading and might face a prison term. Of course, you go to prison for insider trading only if you get caught, and the way not to get caught is to have someone else buy the stock based upon your information. You first consider suggesting to your father that he buy Beta stock. You realize the stupidity of this plan because if the SEC (the stock market police) found out that your father bought the stock, they might become suspicious. You need to get someone with whom you do not have a strong connection to buy the stock. You therefore propose to a friend you haven't seen since high school that he buy lots of Beta stock and split the profits with you. You and your friend have now entered into a trust game. Of course, the safe course of action would be not to engage in the illegal insider-trading scheme at all. If you engage in the insider-trading scheme you have very little chance of getting caught if neither of you does something stupid like brag about your exploits to a friend. If, however, the two of you carry out your plan, you each could suffer a massive loss if your fellow player proves untrustworthy. Workers who go on strike are also often engaged in trust games. Frequently, if all the workers show solidarity and remain on strike until their demands are met, then management usually will have to give in, and the workers will be better off. Going on strike is risky, though, for you will be hurt if your fellow workers abandon the strike before management submits to the demands. The obvious solution to any trust game might be for all parties simply to embark on the risky course. A small amount of doubt, however, might make the risk unbearable. Doubts within Doubts: A Confusing Recursive Discourse Doubts are deadly in trust games. Indeed, even doubts about doubts can cause trouble. For example, assume that both you and your friend have the temperament to successfully execute an insider-trading scheme. Your friend, however, falsely believes that you are a blabbermouth who would discuss your schemes with casual friends. Your friend would consequently not engage in insider trading with you even though you both are trustworthy. Figure 21 shows the mayhem unleashed by doubts within doubts. Obviously, in the game in Figure 21 the best outcome would occur if both people played A. Small doubts, however, could make this outcome impossible to achieve. To see this, imagine that there are two kinds of people, sane and crazy. Assume that a crazy person in the game in Figure 21 will always play B. A rational person will play whatever strategy gives him the highest payoff. Let's assume (perhaps unrealistically) that you are sane. If you think your opponent is crazy, then obviously you will play B and not get the high payoff. Let's assume, however, that you know your opponent is rational, but you also know that he mistakenly believes you to be crazy. If he thinks you are crazy, he will play B and thus you should too. Even if you are rational and your opponent is also rational, the only reasonable outcome, if your opponent thinks you are crazy, is for both of you to play B. Even though both parties are rational and believe the other person too is rational, if you doubt that they know you are rational, you should play B. This, of course, means that if your opponent believes that you believe that your opponent believes that you're crazy, then he should play B and consequently so should you. Thus, to achieve the ideal outcome in a trust game it's not sufficient for both people to trust each other. There must also be an infinite chain of trust where you trust that they trust that you trust that . . . that you're trustworthy. Figure 21 Outguessing Games The opposite of a coordination game is an outguessing game, the classic example of which is the matching pennies game shown in Figure 22. In the matching pennies game two players simultaneously choose either heads or tails. Each separately writes down his choice. If both players make the same choice, then Player One wins, while if they choose different sides, then Player Two wins. [...]... Fixed-Sum Games You can characterize games by their total payoff In a fixed-sum game, what one player gets, another loses Mathematically, in a fixed-sum game, the sum of the players' payoff in each of the boxes adds up to the same number Chess is a fixed-sum game in which if one player wins, the other must lose Variable-sum games consist simply of all games that are not fixed sum In variable-sum games. .. stayed in Interestingly, each firm might welcome attempts by the other firm to spy on them Spying in Chicken Games As you recall, in outguessing games you want to stop your opponent from spying on you In outguessing games, if your opponent figures out your strategy, he always wins In chicken games, however, if your opponent knows your strategy, and that strategy is one of machismo, then you will triumph... you could be a wimp Since perception is everything in chicken games, being perceived as a wimp will indeed make you wimpy It is therefore vital in games of chicken never to openly stop your opponent from watching your actions The difference between counter-spying strategies in chicken and outguessing games comes about because in outguessing games a party always has something to hide Regardless of what... strategy In outguessing games you want to hide your moves When approaching an intersection at night, it would be a very bad idea to turn off your lights to keep the other drivers from guessing your intentions In outguessing games, however, you always want to turn your lights off This is because while in coordination games you are better off if your opponent knows your move, in outguessing games you want to... variable sum? Do I want my opponent to guess my future moves? Am I better off being perceived as rational or crazy? Mass Coordination Games In the next chapter we will look at simultaneous coordination games that are played by millions of consumers The outcome of these games often determines the fate of high technology companies Lessons Learned A dominant strategy gives you a higher payoff than all other... your actions from your opponent Or even better, you want to spread false information Negotiations are pointless in outguessing games since both players have strong incentives to lie The key to winning outguessing games is to hide, never trust, and always strive to deceive Games of Chicken In the classic game of chicken, shown in Figure 24, two cars drive straight toward each other The first driver... player plans to make in an outguessing game, she doesn't want her opponent to become aware of her actions Attempts at concealment in outguessing games tell your opponent nothing, since everyone would always try to hide everything in these games In chicken games, however, a party wants to hide something only when she has a weak commitment to adopting the macho strategy Thus, you can't completely hide... this example shows, you can trust your business partners not to exploit you only so long as your partner still cares about his reputation Free Rider and Chicken Games[ 8] Free rider problems can cause chicken games to manifest In free rider games players try to be lazy and benefit from the efforts of others Consider the game in Figure 26 Imagine that this game came about because a boss assigned two... games the size of the pie varies, so there is room for both cooperation and competition The players have an incentive to cooperate to make the total pie as large as possible and to compete to maximize what percentage of the pie they each get In fixed-sum games outside forces fix the size of the pie, and the players fight over how much of the pie each of them gets to eat Consequently, in fixed-sum games. .. Conversely, the company wants to watch only when the employee steals Remember, in simultaneous games both sides move at the same time and must make their move while still ignorant of what their opponent will do Thus, each side must predict what the other side is up to Figure 23 Randomness manifests in all outguessing games To see this in our stealing game, assume, falsely, that there is no randomness . Challenging Simultaneous Games Games involving dominant or strictly stupid strategies are usually easy to solve, so we will now consider more challenging games. . key to succeeding in coordination games is to be open, honest, and trusting. Trust Games Trust games are like coordination games except that you have a safe