## How to Buy Christmas Presents with Nash Equilibrium

As winter approaches in Ithaca and students become more anxious about their final exams, the season of gifts approaches parents and perhaps some students. But there is always a question for at least one person they know of: do I buy presents for them? In the article “How to apply Game Theory to Buying your Christmas Presents,” the author references one of the episodes in “The Big Bang Theory” to introduce the conflict and propose a solution. If we have a problem buying gifts like Sheldon, then we can apply Game Theory to help us decide.

The article first simplifies the problem by quantifying two variables E (enjoyment) and C (cost). Having two players in the game, we can have the following table represent the problem:

Each of the cells represent the payoffs for players 1 and 2 respectively, but if E > C for both players, then there is a prisoner’s dilemma. If we apply our knowledge in class regarding Nash Equilibrium, then it is best for both players to not give any presents to each other because both of their dominant strategies are to Receive rather than Give. Obviously, this is a simple problem that we students can solve, but the article digs a little deeper to introduce a more complex problem. If we are playing against someone who we will repeatedly spend Christmases with and who will remember our past presents, we are faced with the Iterated Prisoner’s Dilemma. The solution, proposed by Robert Axelrod in his book The __Evolution of Cooperation__, tends to be better for generous rather than greedy strategies. Remembering previous results and actions of the opponents lead to trust and cooperative behavior, and so people become more inclined to give each other gifts. This completely makes sense because as we understand one another better, we can have more enjoyment in giving and also find each other’s preferences of gifts, which can increase the payoff of E_{1} – C_{1 }and E_{2 }– C_{2.}. Now obviously, people have difficulties deciding what others would enjoy even if they become closer. “For small children, for example, there is usually huge enjoyment in whatever they receive and that can be achieved for a relatively small cost. Teenagers are more tricky, almost everything they want is more expensive and although they might enjoy them the joy of receiving is diminished – if only because the size and quantity of the presents is much smaller” (Norman, O’Hare). The article adds even more complexity to the problem by introducing another variable, the joy of giving J. The strategy replaces -C with J-C, and if this new value is always set to be positive, then the dominant strategy becomes always giving gifts.

The article concludes with the idea of generous giving and Sheldon’s ultimate decision. After Sheldon buys several gift baskets of different values to give the one closest in value to Penny’s gift, Penny brings him “the DNA of Leonard Nimoy.” There is no monetary value to this gift, but the enjoyment was so great to Sheldon that he gave her all the gift baskets. Although there seems to be a solution to the problems presented in the article, we must consider cases where the problem is limited. We cannot generally assume that E > C for the first situation and assume that J > C for the second. There are so many factors that change these values, such as the relationship between the players, age difference, personal financial situations, and number of gift exchanges for each player. While reducing the problem with Nash equilibrium might provide a solution to the gift-giving season, numbers cannot, and should not, represent the intangible values a present brings to an individual.

Sources

- “How to apply Game Theory to buying your Christmas presents” by Rachel Norman & Anthony O’Hare: https://theconversation.com/how-to-apply-game-theory-to-buying-your-christmas-presents-52233