Christmas-time Game Theory

For the last four or five years, I’ve made a habit of making Christmas lists. Sure, it takes a little bit of the fun out of not knowing what you’re going to get for Christmas - but you could easily argue that the opportunity cost of that is outweighed by the decreased likelihood that you’ll get clothes for Christmas. This year, however, it might be time for a change. Christmas lists are so tired and boring… I thought it’d be more fun to be provocative. Therefore, in this article, I use the principles of game theory to show that in a group of two people (extensible to groups of size n), the group as a whole is better off when each person buys each other person a large present, as opposed to a small present - and I employ paralepsis to imply what I’d like for Christmas without daring to write something as tawdry as an actual list.

The gift-giving experience can be modeled as what is known in game theory as a “2x2 game”. That is, there are two “players”, representing the two people exchanging gifts (we ignore the case of gifts that are not reciprocated). Each of these players will have a given preference for gifts, but we assume that these preferences are communicated beforehand, and that each player has two gift options to give to the other player - a “large gift” and a “small gift”. To be able to analyse these choices properly, we must assign a value to each of them. The value may have no reflection on the actual dollar cost of the product; gifts with a low cost may still be valued highly by the player receiving the gift. The value is simply a measure of how much each gift is desired. For the purposes of this analysis, we assign a value of 5 “units” to the large gift, and 2 units to the small gift. (It will be shown later that within certain limits, this analysis is independent of the values chosen.) To add some realism to this simple model, we add a cost to the large gift. Shopping for a particularly large item expends time, and that time has its own opportunity cost, which incurrs to the person buying the large gift (as opposed to buying an average gift which might require less shopping time). In this analysis, we ascribe a value of 1 unit to the large gift’s opportunity cost; this value is subtracted from the purchaser’s payoff from the game. Each player now has two choices in their game - to give a large gift, or an small gift. Each combination of choices has a payoff to each player, and the four possible combinations are as follows (I would draw a table, but I’m not an HTML programmer, I’m just an academic):

  1. Player 1 buys a large gift, player 2 buys a large gift: Both players pay 1 unit to buy the good gift, but receive a gift worth 5 units. Both players gain 4 units;

  2. Player 1 buys a large gift, player 2 buys a small gift: Player 1 pays 1 unit for the large gift, and only receives a gift worth 2 units - their net gain is 1 unit. Player 2 pays nothing for their gift, and receives a gift worth 5 units. This is the worst outcome for player 1, and the best outcome for player 2;

  3. Player 1 buys a small gift, player 2 buys a large gift: This is the polar opposite of the above outcome; the best for player 1, and the worst for player 2;

  4. Player 1 buys a small gift, player 2 buys a small gift: Both players receive a gift worth 2 units, and incurred no costs purchasing their gift.

Each player will try to pursue the strategy that gives them the highest payoff. We can look at it from player 1’s point of view (player 2’s view will be identical):

  • If player 2 buys a small gift for player 1, player 1 should not spend 1 unit to buy a large gift for player 2. The best strategy is to buy a small gift.

  • If player 2 buys a large gift, again, player 1 should prefer to buy a small gift, to avoid the 1-unit cost of the large gift.

Both players will follow their best strategy, which is to buy a small gift. Unfortunately, this leads to a result that is suboptimal. Looking at the list of outcomes, we can see that both players would do better to buy each other a large gift. Unfortunately, as soon as one player reneges on the large-gift pact and buys a small gift (although, to be fair, small does not necessarily equate to bad, as the linked counterexample demonstrates), the pact must break down, because the other player will choose to follow their best option and renege as well. This game is not solvable in a way that benefits both parties. (For people who are familiar with game theory, this situation may look pleasingly familiar - and so it should. This example is a Prisoners’ Dilemma, one of only two known classes of 2x2 games that cannot be solved.) Having implied that this game cannot be solved, the next step is to solve the game. The linked document explains that for this version of the Prisoners’ Dilemma, Robert Axelrod conducted an extensive computer simulation and found that the most effective strategy was this:

  • On the first turn (the first Christmas), one should always cooperate (buy a large gift) instead of defecting (buying a small gift).

  • On subsequent turns, one should do exactly what the opponent did on their previous turn. Eventually, it is hypothesised, an intelligent opponent will see this strategy and cooperate on all future turns, maximising the payoff to both parties.

The weightings given to the various gifts can be changed as much as one likes, with one exception - when the value of the large gift drops too far, or the opportunity cost of buying it goes up beyond its value, there is no longer any incentive at all to buy the large gift. If the opportunity cost attached to the large gift drops to zero, then there is no longer any incentive to renege on buying a large gift, because no time will be saved by buying a small gift. This degenerate form of the game is easily solved by having everyone buying large gifts - but the upshot of the empirical solution to a Prisoners’ Dilemma is the same. The best strategy in the first round of the game is to buy large gifts for everybody, and rely on them to “do unto others as they do do unto you”. As to whether or not I’m going to reveal my strategy for this year’s Christmasers’ Dilemma, that would violate the rules of the game. If I announced my strategy, then I’d expect you to immediately follow your best option given the choice I have announced - which is to buy a small present. Clearly, this is not an optimal strategy for me. Hence, I choose to withhold my strategy, but if you’d like to quantify the values you’d attach to certain choices of present, drop me an email at josh (at) thinkshiny (dot) com. Have a very mathy Christmas, although I’ll probably write again before then.