Chapter 3

Standard Game Types

By now you might have started to pick up some patterns in the games we looked at in Chapter 2. There are certain types of common game structures that we see, depending on the relationships between the payoffs in the payoff matrix. Let us discuss a few of these typical game types. Once you learn to recognize them, you will be able to quickly understand how the players are likely to behave.

3.1 The Prisoner’s Dilemma

The most famous type of game is called the “Prisoner’s Dilemma.” It is based on the common premise of many police dramas on television; however, it also occurs in real life. Two people are arrested and interrogated in separate rooms. They are suspected of having committed a major crime, say armed robbery, for which the police do not have strong evidence against them. However, when they were arrested they were found to have a small quantity of illegal drugs in their pockets, a more minor offense. The police explain the probable sentences for each of four possible outcomes to each of the suspects (see Table 3.1):

Here the negative numbers represent “Years of your life lost in jail.” Here, to Rat means to confess to the crime, but agree to testify against your partner, saying that (s)he was the real criminal and that you were just riding along as a minor accomplice. The police will reward you by letting you out of jail today if you agree to testify, but the other person stays quiet, denying responsibility (and going to jail for a long time). If you both deny responsibility, they will put you both in jail for 1 year for the drug charge. However, if you both sign confessions, they now have strong evidence against both suspects; and while cutting them both a deal on the major charge, both will go to jail for 10 years.

There is one Nash equilibrium to this game that is also a dominant strategy equilibrium: They will both Rat each other out and both go to jail for 10 years. No matter what the other player does, both players have a strong incentive to Rat the other player out. For example, whether Clyde Rats or Denies, Bonnie will spend less time in jail when she Rats. However, rather than both spending 10 years in jail they could both spend one year in jail if somehow they could agree to both Deny. However, in a one-shot game it is not reasonable to assume that this will happen.

Thus, the pattern of the Prisoner’s Dilemma is that while there is an option for cooperation that is better, the strong prediction (because both players have dominant strategies) is that they will end up at a much worse outcome. Our game in the previous section about Jack and Joe and the Happy Hours is another example of a Prisoner’s Dilemma. Without some way to enforce cooperation, people in the Prisoner’s Dilemma will end up in a sad situation in the outcome. In Chapter 6, we will explore some ways in which cooperation can develop when games are repeated. At the end of this chapter we will discuss some “game-changing” techniques that can improve outcomes for one or both players.

3.2 Chickens, Hawks, and Doves

There is a dangerous game played by some young scoundrels (at least in the movies). They call it “Chicken.” You need two cars (or tractors in some movies), a long, straight road, and two teenagers without much sense. The two vehicles start at opposite ends of the road and are expected to drive very fast toward each other. If only one of them swerves, then he is the “Chicken.” The “winner” gets to brag and be a big man, while the “Chicken” gets made fun of. If they both swerve, both will lose a little pride. If neither swerves, they will both die. How to represent this kind of payoff depends somewhat on your religious outlook, but let us suppose that it is –100 for the purposes at hand. In the 1984 movie “Footloose,” the characters Ren and Chuck play Chicken on tractors. Suppose their payoffs were the following (see Table 3.2):

In this game, there are two Nash equilibria, where one person swerves and one person goes straight. And, in this game, of course each player wants to be the one that goes straight. Because there are two Nash equilibria it is difficult to predict what these players should do. This is another game where the mode of play will likely be some sort of randomization—either literally, or by there being a certain percentage of people who are reckless and will go straight, and a certain percentage of people who are prudent and will swerve. However, perhaps there are some things that can be done to change this game. Think about how you could make sure that you’re the one that goes straight, while minimizing the chance that the other player also goes straight. We will come back to this in ­Section 3.5. What is needed is a way to try to coordinate the actions of the players. Sometimes people call this kind of game a “conflicting interests coordination” game. Coordinating will help keep people from dying, but the ­players will not be able to agree on who should go straight and who should swerve.

A common variation of this game (that is strategically very similar) is called the “Hawk–Dove Game.” It is used in evolutionary biology to discuss whether aggressive (hawk) or passive (dove) behavior is optimal in an animal population. It turns out that in many cases it is optimal for there to be a mix of some aggressive members of the species and some submissive members of the species, the proportion of each being determined by the payoffs. For example, suppose that there are 10 units of food on the ground. If both birds act as doves they can split the food equally. If only one is aggressive, the hawk gets the food and the dove flies away with nothing. However, if they are both aggressive and fight, they both get seriously hurt (see Table 3.3).

This is very similar to the game of Chicken. There are two Nash equilibria where one plays the role of the aggressor and the other retreats. Unless there is some way to coordinate who should play the hawk and who should play the dove, then we predict that people will probabilistically choose, or there will naturally arise a certain percentage of people or animals in a population that act as aggressors. What we can learn from this sort of game is that a population of only aggressors is certain to die out; in a population of all doves, a lone hawk would have a field day.

However, another interesting type of outcome is that societal rules for behavior develop. For example, a society of people or animals might develop a rule saying that if such a confrontation occurs near your house, and the other player is an outsider, then you are obliged to be aggressive and they are obliged to be submissive. If everyone understands that this is the rule, then we will never need to see the “bad” outcome where the aggressors get into a fight.

One last example of this type of game that is not quite as adversarial is called the “Battle of the Sexes,” though I prefer to call it the “Dating Game.” Bob and Suzy are in love and would like to be together; however, Bob prefers wrestling and Suzy prefers the opera (see Table 3.4).

Just as in Chicken and the Hawk–Dove Game, in the Dating Game there are two Nash equilibria. And, just like in those games Bob and Suzy could use some help with coordination. Another similarity is that Bob and Suzy will not be able to easily agree on which event they should go to. Of course, in a repeated game setting they could alternate to be fair, but in a one-shot game it is unclear what would happen without communication. Even with communication, there is conflict.

3.3 Cooperative Coordination Games

In the previous section, we discussed the need for coordination to avoid two people both being aggressive to their detriment. In a more cooperative type of coordination game, it is in both players’ best interests to be able to communicate and coordinate.

A truly classic example of a coordination game was discussed by ­Jean-Jacques Rousseau where two hunters with spears in the woods (without communicating because they must be quiet) have to simultaneously decide whether to hunt the Stag (male deer) that is between them, or to individually spear a hare (rabbit) that happens to be standing beside each hunter. In order for them to take down the Stag (that would mean quite a lot of food for them) they must work cooperatively; however, the small, tasty rabbits are a sure thing. The payoff matrix appears in Table 3.5.

As in the Dating Game and the Hawk–Dove Game, there is a way for cooperation and coordination to improve outcomes. Like the Hawk–Dove Game, there are two Nash equilibria; they should either both hunt the Stag or both hunt their individual rabbits. Hunting the Stag is risky, because getting any food depends on the other person cooperating, while getting the rabbit is safe, yet with a lower payoff. The outcome where both hunt the Stag makes more sense, but does require a level of trust. Whether these two hunters have any sort of belief system or “trust” that makes them think that the other hunter will join them in the Stag hunt has important implications for how successful these hunters will be in the long run. This kind of cooperative behavior is seen in animals like lions and Orca whales, and some people suggest that there is an innate trust mechanism among humans, at least among members of the same group.

A slightly different example is what we call a “pure” coordination game. In this kind of game, it matters that the two players coordinate in the correct way, but it does not matter exactly how they do it. In the game below, people have to decide which side of the road they’re going to drive on (see Table 3.6).

As long as everyone decides to drive on the right side of the road, or everyone decides to drive on the left, there is no difficulty. Neither player has a preference, but it is extremely important that coordination does occur.

3.4 The Zero-Sum or Fixed-Sum Game

In a Zero-Sum or a Fixed-Sum Game, the only way one player can ben­efit is at the expense of another. In this type of game, the interests of each player are directly opposed. These games are sometimes called “strictly competitive” because there is absolutely no room for cooperation here. An example is a simple game called “matching pennies.” Grab a large jar of coins, and divide them equally among two players. In each round of the game, players must simultaneously place a coin down on the table with either heads or tails up. One player is called the “evens” player and the other is called the “odds” player. If the pennies match then the evens player wins the other player’s penny. This game is depicted in Table 3.7(a).

The outcome and intuition for this game is straightforward. Neither player has a dominant strategy; that is, the evens player does not always want to play heads nor does he always want to play tails. The evens player wants the pennies to match so he wants to play heads when the other player plays heads, and tails when the other player plays tails. However, the odds player does not want the coins to match. In this game there will always be one player who is happy and one who regrets their decision, as in each of the four possible outcomes there is always one winner and one loser. In this game, the best way to play is to randomize; you might as well flip a coin in the air rather than actually make a choice. There is a “Mixed Strategy Nash Equilibrium,” where each player chooses “heads” 50% of the time. Any other probabilities can be taken advantage of by the other player to win more than one-half the time. For example, if the odds player played “heads” 60% of the time, once the evens player ­figured this out, he could win more than one-half the time by also ­playing “heads” frequently.

We call this game a Zero-Sum Game because the payoffs in each of the four boxes add up to zero. The payoffs don’t have to be –1 and 1, they could just as easily be –6 and 6, as long as they sum to zero. Actually, the behavior in a game like this will remain unchanged even if the payoff in all outcomes sum to any number (called a Constant Sum Game). Look at Table 3.7(b). Here, two players are playing Matching Pennies with an additional rule. At the end of each round, each player must throw a penny into a well. So, the winner takes the penny he won and throws it away, while the loser not only loses a penny in the game, he loses an additional one to the well.

However, Zero- and Constant-Sum Games sometimes do have clear outcomes. The game described next is also a Zero-Sum Game. ­However, in this one we have a clear prediction of what the outcome will be (Table 3.8):

Player A has a dominant strategy to always choose “up.” Player B does not have a dominant strategy; he would rather choose “left” if Player A chose “down” but “right” if Player A chose “up.” However, as Player B knows that Player A will always choose “up,” he minimizes his losses by choosing “right.” This is a Nash equilibrium of the game as neither player has an incentive to change their minds from this outcome, and this is the only outcome that makes sense. In the next chapter, we will look at some larger games with more choices for each player, and also discuss some “refinements” to the ideas of dominant strategies and Nash equilibria that can help clarify our predictions of a game’s outcome.

3.5 Thinking Outside the Box: Change the Game You’re Playing

Now that we have discussed a wide variety of games and the importance of all players understanding what the true payoffs are, let us think about some practical lessons that will be valuable in the real world when we encounter games such as those that we have seen. When investigating simultaneous games, the starting point is to assume that players have to make their choices at the same time without any communication. We also assume that they have to play the game that they’re given, and have no way to change it. In the real world, however, we normally have the opportunity to alter the rules, alter the payoffs, communicate, or change the game in other ways.

Prisoner’s Dilemma

If you find yourself in a Prisoner’s Dilemma, you want to think about ways to ensure that the other player (or if you are benevolent, both of you) choose the dominated strategy. It is important to note that communication alone will not help the situation. Even if both players “promise” to ­cooperate, the fact that they have a dominant strategy to rat the other player out will overrule any empty promises. Also interesting is that in the Prisoner’s Dilemma, you cannot change the other player’s mind to ­cooperate even if somehow you can convince him that you really will cooperate. Because it is in your opponent’s best interest to rat you out regardless of what he thinks you’re going to do, even finding a way to commit yourself to cooperating will not change his mind. An external force is needed to achieve cooperation. Some drug gang members are “encouraged” not to testify against each other by an outside force: other members of the gang that might be either inside or outside of the prison. Someone who testifies against another gang member could be punished either inside or outside prison themselves, or have life made difficult for their families outside of prison. We saw a Prisoner’s Dilemma type game in Chapter 2 where Joe’s and Jack’s bars had dominant strategies to have Happy Hours, which actually hurt profits when both engaged in this practice. We discussed how the bar owners could go to an outside authority and ask for Happy hours to be made illegal, pretending to do this in the public interest. Perhaps they could say that Happy Hours encourage people to drink a lot of cheap alcohol in a short period of time and it is thus unhealthy.

The extraction of natural resources is another example of a Prisoner’s Dilemma: Whether it is extraction of oil or fish from the ocean, individual players have an incentive to overharvest for short-term gain, at the expense of long-run survival and profits. Major oil extractors founded an organization called OPEC to try to come up with agreements to achieve the cooperative outcome. Governments and fishermen have tried to achieve similar results through agreements and regulations, which sometimes work well.

In Chapter 6, we will discuss how repeating a game can, in some circumstances, lead people in a Prisoner’s Dilemma to cooperate. It is important to note that cooperation does frequently occur in repeated games, so we should not take away the idea that selfishness is always the rule. Elinor Ostrom, the 2009 Nobel Laureate (along with Oliver Williamson) gained her fame by describing real-world situations where people figured out ways of cooperating amongst themselves when faced with Prisoner’s Dilemma-type games. She found that this worked well when members of a community can set up and enforce their own rules, as long as monitoring and punishing “bad” behavior is possible.

Chicken and Hawk–Dove Games

In a game of Chicken or the Hawk–Dove Game, your goal is to convince the other player that you will be the Hawk and will not and cannot back down. Of course, the other player has this in mind as well. Two strategies that might be worth considering here are “tying your own hands,” or changing the payoffs (or at least convincing the other player that your payoffs have changed).

“Tying your hands” means figuring out a way to eliminate one of your choices. For example, at the beginning of a game of Chicken you could rip off the steering wheel and throw it out the window of the car. Thus, you have committed to going straight and cannot swerve. After you do this the game will look like Table 3.9.

Now that your opponent knows that you must go straight, they have a choice between going straight for a payoff of –100, or swerving for a payoff of –2.

Another viable strategy is to make your opponent believe that your payoffs are different than the normal person’s payoffs. There are two ways that you could do this:

1. Convince your opponent that for you to swerve would bring so much dishonor that it would be worse than death. Convince them that you would have to kill yourself anyway, and that it would also dishonor your family (Table 3.10(a)).

2. Convince your opponent that you have absolutely no fear of death or bodily harm (Table 3.10(b)).

Japanese Samurai warriors cultivated both kinds of reputation to some extent in their code of “Bushidô,” or “the way of the warrior.” By having the reputation for being fearless, and of cowardice being worse than death, they gained an advantage over opponents without such a reputation. In either case, if you are successful at developing this kind of reputation you will have convinced your opponent that you have a dominant strategy for taking the straight path. The only rational move left for them is to swerve.

The Dating Game

In the Dating Game, slightly different ways of commitment or tying your hands could be tried. Bob could buy tickets to the wrestling match. Or an interesting suggestion by Nobel Prize winner Thomas Schelling could be used: Deliver one message and then cut off communication. This is done in movies or TV sometimes. Bob could call Suzy and say “I’m on my way to the wrestling match, and my cell phone battery is about to die… <click>.”

Cooperative Coordination Games

In these games communication is really all that is necessary, absent some sense that the other player is “out to get you.” The payoffs put it in each other’s best interests to stick to any agreement that is made, and there is no reason for either player to try to attempt to “renegotiate” as there is in the Chicken, Hawk–Dove, and Dating Games.

Zero-Sum and Fixed-Sum Games

Even though the phrase “Zero-Sum Game” is a bit overused to describe real-world situations, they are actually not all that common. Many times, people erroneously describe games such as the Hawk–Dove Game as a Zero-Sum or Fixed-Sum Game because they are fighting over a fixed amount of food, but ignore the fact that resources are wasted in the fight for it.

In a Zero-Sum Game, where there is no “pure strategy” Nash equilibrium, the advice is to truly randomize. However, you can always come out ahead if there is some way to force your opponent to go first. If you can trick your opponent into revealing their choice (say, heads in matching pennies), then you can always optimally respond and win. However, since this type of game is more or less equivalent to gambling, perhaps it is best not to play (except for fun!).