Game Theory John R. Swinton, Ph.D. Center for Economic Education Georgia College & State University

Game Theory Question Take from 2013 AP Free Response Section: There are two pizza restaurants in College Town, Piecrust and LaPizza. Each company must decide whether to advertise or to not advertise. In the payoff matrix below, the first entry in each cell indicates PieCrust’s daily profit, and the second entry indicates LaPizza’s daily profit. Both firms have complete information. Lapizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 I am going to work my way through one of the more recent game theory questions asked in the AP Micro exam. While the answer itself is fairly simple, what makes it simple is an understanding of what the game theory presentation tells you. So as I work my way through the problem I will try to decode the various questions.

Game Theory Why Game Theory? When markets are neither perfectly competitive nor monopolistic models become more complicated Agents must consider reactions to their actions (strategy) Game theory provides a framework to help study strategy First, game theory is just another tool economists use to analyze market behavior. The perfectly competitive and monopoly models have fairly straightforward implications if all of the underlying assumptions hold. But, in many markets there are not either a sufficiently large number of buyers and sellers to assume perfect competition or only one seller (or one buyer) to assume a monopoly (or monopsony). The area in between these two market structures is host to a wide array of possible types of interactions. One area of interest is that of strategic interactions. We assume agents have an objective – such as maximizing profits. We assume that they will rationally set out to achieve that objective. We also assume that they will have to interact with another agent who has similar objectives. Game theory provides a way to assess the value of different reactions individuals will have when they choose to interact in different ways.

Game Theory Elements of the game Players Lapizza Piecrust Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 Elements: Every game is defined by who is a player in the game – that is who is allowed to make a decision in the game. At the introductory level it is typical to have two players in the game. But games can have many more players. It is also possible to introduce chance into the game is some results are subject to some random element. In this case “Nature” becomes a “player” in the game and represents the random element. In this problem our two players are Piecrust (which I will represent with the color green) and LaPizza (which I will represent with the color red).

Game Theory Elements of the game Strategies Advertise Not Advertise LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 Elements: Every game is also defined by the strategies that players may choose. Again, simple games often limit the choices to two options. But, there is no theoretical limit to the number of strategies available. The key idea is that each strategy one player chooses will have a payout associated with it that depends both on that player’s strategy and the strategy that the other player(s) chooses. Notice that in this case, if player Lapizza chooses to advertise its payout will depend on whether Piecrust chooses to advertise or not advertise. The same is try for Piecrust.

Game Theory Elements of the game Payouts Lapizza -- $200, $300, $500, or $400 Piecrust -- $250, $180, $450 or $390 LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 Elements: Every game is also defined by the strategies that players may choose. Again, simple games often limit the choices to two options. But, there is no theoretical limit to the number of strategies available. The key idea is that each strategy one player chooses will have a payout associated with it that depends both on that player’s strategy and the strategy that the other player(s) chooses. Notice that in this case, if player Lapizza chooses to advertise its payout will depend on whether Piecrust chooses to advertise or not advertise. The same is try for Piecrust.

Game Theory Elements of the game Presentation Strategic form Extensive form LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 Elements: Games can be presented in a variety of ways. The two most common are Strategic form such as the one we are working with. The other is Extensive form (following slide). The strategic form is preferable for games where both players have to move (enact their strategy) at the same time. Games like this include rock-paper-scissors. We refer to these games as simultaneous games.

Game Theory Elements of the game Presentation Strategic form Extensive form Elements: Here is a picture of a simple extensive form game. It represents a game where players move one at a time such as tic-tac-toe. Player 1 moves first and can choose strategy up (U) or down (D). Player 2 responds with either U’ of D’ . The final node of the tree indicates the payouts that result from the various strategies.

Game Theory Types of Games Cooperative vs. Non-cooperative Simultaneous vs. Sequential Zero Sum vs. Positive Sum Single Play vs. Repeated There are many types of games that game theorists have modeled: Cooperative (Cartel) vs. Non-cooperative (Most games) Simultaneous (rock-paper-scissors) vs. Sequential (Tic-Tac-Toe) Zero Sum (Rock-Paper-Scissors) vs. Positive Sum (Trade) Single Play vs. Repeated (Reputation) Bring up Crazy Eddie for Mike

Game Theory (a) What strategy should PieCrust choose if LaPizza chooses to advertise? Explain using the dollar value in the payoff matrix. Payout for Advertise = $250 Payout for Not Advertise = $180 LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 Back to the question at hand: The first part asks us to examine the results of different strategies. Let’s examine what is available if LaPizza chooses to advertise. What is Piecrust’s payout if it chooses to advertise? $250. What is Piecrust’s payout if it chooses not to advertise? $180. Which is the better option (assuming it wants to maximize profits)? To advertise.

Game Theory (b) What is the dominant strategy, if any, for LaPizza? Explain using the dollar values in the payoff matrix. None for LaPizza Advertise for Piecrust LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 There are strategies and then there are better strategies. What is a dominant strategy? A dominant strategy is one that is better than other strategies no matter what the other players do. So, let’s compare LaPizza’s two strategies and see if one is always better than the other. Remember, we are looking at the second entry in each cell. If Piecrust chooses to advertise, then LaPizza will get $200 if it advertises and $300 if it does not advertise – Not Advertise is the better strategy. If Piecrust chooses to not advertise, then LaPizza will get $500 if it Advertises and $400 if it does not Advertise – Advertise is the better strategy. In this case, LaPizza does not have a dominant strategy. Let’s turn it around and look at Piecrust. If LaPizza chooses to advertise, then Piecrust will earn $250 if it Advertises and $180 if it chooses to Not advertise – Advertise is its better strategy. If LaPizza chooses to Not Advertise, then Piecrust will earn $450 if it Advertises and $390 if it chooses to Not advertise – Again, Advertise is the better strategy. Therefore, Piecrust will choose the Advertise strategy no matter what LaPizza does and it is the dominant strategy.

Game Theory (c) In the Nash equilibrium, determine each of the following. i. PieCrust’s daily profit ii. LaPizza’s daily profit LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $180, $500 $390, $400 The Nash equilibrium is a particular type of equilibrium that is slightly different than what students will be accustomed to by this point in an economics class. The notion that Nash introduced is that given a game of strategy, there will be certain strategies that, once adopted, will produce outcomes that will be resistant to change. Put another way, given the strategy of the opposing player, players would not choose to change their strategy. To see how this works, pick a strategy for LaPizza – Advertise, for example. Given that LaPizza advertises the best response for Piecrust would be to Advertise. But, if Piecrust were to advertise the best strategy for LaPizza would be to Not Advertise. Now if LaPizza were to Not Advertise the best strategy for Piecrust is to stay at Advertise. Therefore the joint strategy of Piecruse Advertising and LaPizza Not Advertising is stable and a Nash Equilibrium.

Game Theory (c) In the Nash equilibrium, determine each of the following. i. PieCrust’s daily profit ii. LaPizza’s daily profit LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $250, $200 $450, $300 $260, $500 $390, $400 It is possible for a game such as this to have multiple Nash equilibria. For example, raise Piecrust’s payout for Not Advertising (when LaPizza Advertises) from $180 to $260. Notice that if Piecrust did Not Advertise and LaPizza Advertised this outcome would also be stable – neither player would benefit from changing their strategy. It is possible to write a payout matrix that does not have any Pure Nash Strategies. A pure strategy is one that a player would choose all the time given what the other player is doing. In those cases (more advanced cases) players can choose to randomize their strategies in systematic ways (such as Advertise 30% of the time and Not Advertise 70% of the time). The beauty of the Nash concept (and what is worth a Nobel Prize) is that Nash showed that every game was at least one equilibria – a concept that is very important in theoretical economics.

Game Theory (d) Suppose that advertising costs increase by $60 per day. Redraw the payoff matrix to reflect the effort of the higher advertising costs. LaPizza Advertise Not Advertise Advertise Piecrust Not Advertise $190, $140 $390, $300 $180, $440 $390, $400 The only task here is to recognize that the act of Advertising costs more and the payout matrix will reflect the added cost. This is to ensure that students recognize the activities that are supporting the strategies and their payouts. There is an interesting change in the Nash Equilibrium that is worth noting. Piecrust’s previous dominant strategy of Advertising is not only weakly dominant – it is indifferent between advertising and not advertising if LaPizza chooses to Not Advertise. The stability of Nash Equilibrium (as a pure strategy) is brought into question. Conceivably Piecrust might fluctuate between advertising and not advertising if LaPizza chooses Not Advertising. But, further analysis shows that all of the other outcomes are unstable assuming no collusion.

Game Theory