Game Theory NotesKyle Anderson
To start, imagine a simple game to determine your grade for the class. I should acknowledge that this example was blatantly ripped (or “borrowed” if you prefer the academic terminology) off from a Yale game theory lecture. The game to determine your grade works as follows. You get to make a decision now that will determine your final grade. You can either choose the “blue” grading scheme or the “red” grading scheme. Once you make your choice, I will randomly pair your choice with that of another classmate in the same section. You will not get to know who that classmate is, either at the time or later on. You will only get your final grade. The grading scheme will be as follows:
- If you both choose the blue grade scheme, you both get an A-.
- If you both choose the red grade scheme, you both get a B.
- If you choose the blue scheme and your matched partner chooses red, you get a B- and your partner gets an A.
- If you choose the red scheme and your matched partner chooses blue, you get an A and your partner gets a B-.
Which grade scheme do you choose?
There are a couple things to notice about this game. First of all, it is a simultaneous move game. Both players make their decisions without being able to see what the other player chooses. Secondly, it is a one-time game – there are no repeated interactions since it is played once and you don’t get to know with whom you were matched up.
So what are the payoffs for each player? First we have to decide how the players value the payoffs. Game theory assumes that players will maximize their payoffs in strategic interactions, however, payoffs are different for different people. For sake of argument, we will assume that each player is only interested in their own grade (and not that of their counterpart), and that each player would prefer a higher grade to a lower grade. These are not innocuous assumptions. There are many instances in which you might be concerned about others payoffs as well. For example, you may just have altruistic feelings of helping out a classmate and therefore prefer that your classmate also do well. Conversely, you may want your classmate to get a low grade so that you have a higher relative grade. But for now, assume that we are only concerned about our own payoff.
Here are the possible outcomes:
Player 2
BlueRed
Player 1BlueB+, B+B-, A
RedA, B-B, B
Since each player has two choices, we can represent the choices in a 2 x 2 matrix that shows the outcomes based on each of the possible choice combinations. By convention, player 1’s choices are on the left while player 2’s choices are across the top. In the matrix, player 1’s payoff is listed first. However, to show preferences, it is often easier to assign numerical values to the possible outcomes. So we will randomly assign the following outcomes: A 4.0, B+ 3.3, B 3.0, and B- 2.7. Again, each player is trying to maximize his/her payoff. Now the payoff matrix looks as follows:
Player 2
BlueRed
Player 1Blue3.3, 3.32.7, 4.0
Red4.0, 2.73.0, 3.0
Given the above payoffs, what do we expect to happen? To find out, let’s look at player 1’s choice. Player 1 should base the decision on the possible choices by player 2. If player 2 chooses Blue, player 1 would prefer to play Red, because this leads to a payoff of 4.0 rather than 3.3. If player 2 chooses Red, player 1 would prefer to play Red, because this leads to a payoff of 3.0 rather than 2.7. Therefore, given the described payoffs, player 1 is always better off choosing Red. In game theory, this is known as a Dominant Strategy – a strategy that is always best regardless of the other player’s choice. Since the game is symmetric, it is not surprising that player 2 also has the same dominant strategy of choosing Red.
Both players choosing red is known as the Nash Equilibrium. A Nash Equilibrium (NE) occurs when each player is doing the best that they can do given the other players’ actions. Another way of putting this is that neither player can do better by switching his/her choice.
One of the first things you should observe about this outcome is that it is sub-optimal from the players’ points of view. That is, both players would be better off if they both chose Blue instead. However, by acting in their own self-interest, they end up at the lesser alternative. This type of game is known as a Prisoners’ Dilemma game. (You should probably understand where the term Prisoners’ Dilemma comes from, but rather than having me recount it for you, it’s easier (for me) to have you Google it and find one of the many abundant answers). Business is full of prisoner’s dilemma games. One type is the Cournot oligopoly game, where each firm overproduces relative to the collusive outcome. As a result, quantity is higher than the monopoly output, and the price is lower. Bertrand competition is another form of Prisoners’ Dilemma where firms consistently try to undercut each other. The result is that price is lower, and in true Bertrand it is equal to marginal cost. Notice that the game is sub-optimal from the players’ points of view, however in the above games, consumers actually benefit from the prisoners’ dilemma game.
The Prisoners’ Dilemma is one type of game. There are a couple things to keep in mind about this particular game. First, it is a simultaneous move game. In this case that is not especially important, as the outcome would be the same even if one player got to make the first move. The second thing to notice is that it is a one-shot game. That is, it is only played one time, and the fact that you don’t know who you are playing against makes it impossible to change the game by bribing or threatening the other player. So it is difficult to change the payoffs.
Game theory is the art (or science) of viewing strategic interactions so that we can make predictions about the optimal strategy and the likely outcomes from the game. There are two general parts. The first part is understanding the game and possible outcomes, and determining strategies that maximize your payoffs. The second part of game theory comes from trying to change the game in such a way that it leads to a higher payoff from the player’s point of view, or at least understanding which games should be played. To understand game theory, it helps to understand the elements. These include:
- Players – by definition a game should have more than one player but generally only “a few.”
- Strategies – each player should have some decision, otherwise they are not really a player.
- Payoffs – each player is assumed to get some payoff based on the decisions of the players.
- Rules – rules govern the timing of strategic moves and communication between parties.
- Information – most games in this class will assume players have complete information about both their own payoffs and that of the other players.
- Rationality – mostly we assume rational players, although we will talk about exceptions to this.
These elements are similar to the games that we think about as games (i.e. sporting events, board games, etc.). One of the key differences between economic games and classic games are that very often classic games are zero-sum games. That is, by definition, if one player does well, the other player(s) are worse off. If we are playing Monopoly and you get $200 for passing Go, I am worse off because that $200 increases the probability that you will win, and thereby increases the probability that I will lose. In the Prisoners’ Dilemma game, all players are better off in the cooperative outcome than the NE. Therefore, a Prisoners’ Dilemma game is not a zero-sum game.
Fortunately, Prisoners’ Dilemma games are not the only type of simultaneous move, one-shot games. Here is another, very simple example:
Player 2
WhiteYellow
Player 1White1, 10, 0
Yellow0, 01, 1
This game is called a coordination game, because both players have an incentive to coordinate their choices to maximize payouts. In this game, there are two NEs – one in which both players choose White and one in which both players choose Yellow. However, given a one-time playing of this game (and assuming they cannot communicate), it is not clear that either NE will be reached. The existence of an NE does not necessarily mean that outcome will be reached, it only means that if that outcome is reached, neither player will want to deviate from it.
This is a variation of the coordination game:
Player 2
WhiteYellow
Player 1White5, 50, 0
Yellow0, 01, 1
There are still two NEs in this game (W,W and Y,Y). We can predict that most players would play white and so the upper left outcome is probably most likely. But recall that the definition of an NE is that no player would change. If the other player is going to play Yellow, I would want to play Yellow as well.
A slightly more devious version of the coordination game is as follows:
Player 2
WhiteYellow
Player 1White5, 10, 0
Yellow0, 01, 5
Each player has an incentive to coordinate, however, the payoffs are asymmetric. Player 1 has a strong preference for coordinating on White, while player 2 has a strong incentive for coordinating on Yellow. Again, there are two NEs, but game theory doesn’t do much for helping us determine which strategy to play or to help us predict which outcome will occur.
There are a couple ways to try to bend the rules or strategically change the game in your favor if you were playing the above game. One is that you might try to “jump the gun” and play first. If player 1 was able to go first, then she would choose White and player 2 would want to also choose white to earn a dollar. However, if this is truly a simultaneous move game, that would not be allowed. A second way would be to come to a collusive agreement where player 1 would pay player 2 some amount to play white so that the winnings were divided. However, in some games, this type of collusion is not allowed. Thirdly, player 1 could announce her intentions and could try to commit to playing white in advance. However, player 2 might try the same game with committing to play yellow, and these announcements may not be credible. A final strategy for player 1 might be to write a contract that would force her to pay an outside party (or even player 2) $20 if she chose yellow. Since she would not want to pay $20, playing White would become a dominant strategy for her – she would play it and player 2 would expect her to play it – and then player 2 would maximize payouts by choosing Yellow.
Business examples of coordination games come in the form of standards wars. For years, the makers of HD-DVD and Blu-Ray fought to become the standard for High Definition DVD. It is in the best interest of the firms to have a single standard so that consumers know what to choose, however, different companies will profit differently from whichever choice they can coordinate on.
For a different type of outcome, try to find the NE of the following game:
Player 2
WhiteYellow
Player 1White1, 00, 1
Yellow0, 11, 0
Player 1 gets a positive payout when the two players match, whereas Player 2 gets a positive payout when they do not match. None of the four boxes in the matrix represents a NE, since one player could always deviate and do better. There is an NE, but it is not in pure strategies. Rather it is what we call a Mixed Strategy Nash Equilibrium. That is, the best strategy for both players is to randomly make a choice of either White or Yellow, but one of the key elements is that the other player must not be able to predict which strategy you will choose. In this case, the NE is for each player to play White with probability ½ and play Yellow with probability ½.
Many sports and games have mixed strategy equilibria. Rock-paper-scissors is one of the simplest. One example from sports is football, where teams generally choose to either run or pass with each play. If the defense knows what is likely to come, they are much more effective at stopping it, so the offense mixes its strategy in part to be unpredictable. Baseball pitchers and batters play a similar guessing game.
Some pricing games involve mixed strategies. Imagine two firms that sell a product, and each firm has some loyal customers. The loyal customers are willing to purchase at their firm as long as the price does not exceed $1. Marginal cost is $0 for each firm, each firm has 25 loyal customers, and there are 50 customers that are “shoppers” who will buy from the lowest priced seller once prices are revealed. Each firm makes its pricing decision at the same time and sends it in to the newspaper where it appears in the respective advertisements for the two firms. How should firms price? This is similar to a Bertrand game except the loyal customers play an important role. Firms have an incentive to undercut their rival, but the undercutting will only go so far, since rather than cutting my price too far, I would rather price high ($1) and make a profit off my loyal customers. In this case, the firm should never go below $.33, since even if it wins the shoppers at that price, it will still make less than it would by charging a dollar and selling only to the loyals. In this case, the optimal strategy is to randomize price between $1 and $.33. While firms rarely actually randomly choose a price (the airline industry has been known to randomize prices – an effective way to prevent firms from consistently matching and/or undercutting you) they often have discounts that may be random (or at least unpredictable). Each week the grocery store puts out a flyer with its weekly specials.