### Super Picks

by Ben Resnick
My friend Kasie and I are both big football fans. We’ve tried to pick the winner in every NFL game since the start of the season, where each correct pick earns one point. With only the Super Bowl remaining, we’re a single point apart. Quite unfortunately, I happen to be trailing by that point, so my chance to avoid an offseason of taunting rests on picking the Super Bowl correctly and having Kasie pick incorrectly. Being an economist, I realized I could apply game theory to guide my strategy for making my final pick of the season in order to maximize my chance of beating Kasie. First, suppose my friend and I only care about if I tie or lose; losing by one point is exactly the same as losing by two. Also, as we have done all season long, our picks will be submitted before the football game starts and then revealed simultaneously.

For starters, assume that both the Green Bay Packers and Pittsburgh Steelers have a 50% chance to win, and further assume that when we make our picks, Kasie and I both know this. The matrix on the right shows the utilities (or payoffs) associated with different combinations of picks. For example, if we both pick the Packers, I cannot gain any ground on Kasie for the season; therefore regardless of the outcome of the Super Bowl, I lose to her and get a utility of -2, while Kasie wins the picking game and receives a utility of 2. However, in the case where we do not pick the same team, no matter who I pick, I have a 50% chance of tying her for the season, and a 50% chance of losing by 2 points. Before the game starts, that gives me a utility of 1 unit, while Kasie gets a utility of -1 because she’ll have to sit through a now stressful game. Therefore, the utilities expressed in the payoff matrix when we play different actions represent ex ante payoffs—that is, they are our expected payoff before knowing the result of the game.

In this case, this is a complete information simultaneous game that has no equilibrium where either player can use a pure strategy. However, there is a mixed strategy Nash Equilibrium for this game where both players randomize their selection and pick either team with a 50% probability. When one player randomizes his or her selection by picking either team half of the time, the other player’s best response is to also pick each team half the time. If both players randomize this way, neither has an incentive to deviate from that strategy, and thus those strategies are an equilibrium.

There is at least one more complication, though, and it’s very important! I am a proud Pittsburgh Steelers fan, so I would prefer to root for my team knowing that I also have a chance to tie Kasie. It would not be as rewarding to root for the Steelers knowing that they must lose in order for me tie for the season. On the other hand, Kasie is a stinky New England Patriots fan, and she has no preference between the teams in the Super Bowl. Because I care about who I want to win, I now prefer not only to pick differently than Kasie but also to pick the Steelers so I can wholeheartedly root for them. The matrix on the left now incorporates my rooting interest by updating my payoff if Kasie and I disagree, while leaving Kasie’s payoffs the same no matter who we both pick. Given the new payoff matrix, the mixed strategy Nash equilibrium is that I pick either team with equal probability, and Kasie picks the Packers 1/3 of the time, and the Steelers 2/3 of the time. If Kasie doesn't adjust her mixing, then I would always pick the Steelers since the expected payoff would be higher than any mixing strategy. My mixing strategy stays the same, however, because Kasie's payoffs are untouched.

Of course, Kasie is going to read this post as well, so now I’m going to need to account for the fact that she knows my preference for the Steelers. Looks like it’s time to update my strategy again!

Discussion Questions:

1. Consider the solution to the picking game when my rooting preferences are factored in. Despite my interest in the Steelers, why do I still only pick them half of the time? If I picked them more than half of the time, what would Kasie’s optimal response be?

2. What is your favorite sports team? Which is more important to you, seeing them win a championship or winning a competition with your friends? How do personal preferences of one player influence the decisions each player makes in this picking game?

3. Suppose one person picking has information about the game that the other does not. For example, if one person gets a tip on an injury to a star player that isn’t public knowledge, how would this new information change the informed picker’s strategy?

4. Some people actually prefer to bet against their favorite team, using the wager as a form of insurance. The logic being that “I won’t mind losing money if my favorite team wins, but if my favorite team loses; at least I’ll get some cash to make me feel better.” How would thinking like this change the way the picking game’s payoffs are described?

For starters, assume that both the Green Bay Packers and Pittsburgh Steelers have a 50% chance to win, and further assume that when we make our picks, Kasie and I both know this. The matrix on the right shows the utilities (or payoffs) associated with different combinations of picks. For example, if we both pick the Packers, I cannot gain any ground on Kasie for the season; therefore regardless of the outcome of the Super Bowl, I lose to her and get a utility of -2, while Kasie wins the picking game and receives a utility of 2. However, in the case where we do not pick the same team, no matter who I pick, I have a 50% chance of tying her for the season, and a 50% chance of losing by 2 points. Before the game starts, that gives me a utility of 1 unit, while Kasie gets a utility of -1 because she’ll have to sit through a now stressful game. Therefore, the utilities expressed in the payoff matrix when we play different actions represent ex ante payoffs—that is, they are our expected payoff before knowing the result of the game.

In this case, this is a complete information simultaneous game that has no equilibrium where either player can use a pure strategy. However, there is a mixed strategy Nash Equilibrium for this game where both players randomize their selection and pick either team with a 50% probability. When one player randomizes his or her selection by picking either team half of the time, the other player’s best response is to also pick each team half the time. If both players randomize this way, neither has an incentive to deviate from that strategy, and thus those strategies are an equilibrium.

There is at least one more complication, though, and it’s very important! I am a proud Pittsburgh Steelers fan, so I would prefer to root for my team knowing that I also have a chance to tie Kasie. It would not be as rewarding to root for the Steelers knowing that they must lose in order for me tie for the season. On the other hand, Kasie is a stinky New England Patriots fan, and she has no preference between the teams in the Super Bowl. Because I care about who I want to win, I now prefer not only to pick differently than Kasie but also to pick the Steelers so I can wholeheartedly root for them. The matrix on the left now incorporates my rooting interest by updating my payoff if Kasie and I disagree, while leaving Kasie’s payoffs the same no matter who we both pick. Given the new payoff matrix, the mixed strategy Nash equilibrium is that I pick either team with equal probability, and Kasie picks the Packers 1/3 of the time, and the Steelers 2/3 of the time. If Kasie doesn't adjust her mixing, then I would always pick the Steelers since the expected payoff would be higher than any mixing strategy. My mixing strategy stays the same, however, because Kasie's payoffs are untouched.

Of course, Kasie is going to read this post as well, so now I’m going to need to account for the fact that she knows my preference for the Steelers. Looks like it’s time to update my strategy again!

Discussion Questions:

1. Consider the solution to the picking game when my rooting preferences are factored in. Despite my interest in the Steelers, why do I still only pick them half of the time? If I picked them more than half of the time, what would Kasie’s optimal response be?

2. What is your favorite sports team? Which is more important to you, seeing them win a championship or winning a competition with your friends? How do personal preferences of one player influence the decisions each player makes in this picking game?

3. Suppose one person picking has information about the game that the other does not. For example, if one person gets a tip on an injury to a star player that isn’t public knowledge, how would this new information change the informed picker’s strategy?

4. Some people actually prefer to bet against their favorite team, using the wager as a form of insurance. The logic being that “I won’t mind losing money if my favorite team wins, but if my favorite team loses; at least I’ll get some cash to make me feel better.” How would thinking like this change the way the picking game’s payoffs are described?

Labels: Game Theory, Mixed Strategies, Nash Equilibrium