What Game Theory Predicts and What It Fails to Predict
In the column, Kominers, the author, invites the readers to play a math game. The rule is as following: everyone submits a number between 0 and 6400 inclusive, and whoever gets closest to four-sixths of the average of all the numbers wins. Based on the reasoning provided in the article, a rational individual should never submit a number over 42,667 (four-sixths of 64,000), since the average will never exceed 64,000. Nevertheless, moving one step further, it is highly probable that others calculate similarly, and all submit numbers under 42,667. Then, in this case, the individual’s best response—or the best range of numbers—isn’t 0-42,667 anymore. Instead, he/she should submit a number less than 28,444 (four-sixths of 42,667) to maximize the chance of winning.
The logic behind the conundrum draws an analogy to the principle of game theory we discussed in class: the game presented involves some players, each with a number of strategies to choose from; players make their own decisions and have no direct access to what numbers others put into the pool; an individual’s payoff depends not only on his/her own choice of strategy, but also on those of others who participate in the game. In this example, there isn’t a dominant strategy for an individual to take. In other words, there isn’t a specific number that can guarantee a win for the player. Besides, the game is also unlikely to have a pure strategy Nash equilibrium due to its nature of high dependency. As is illustrated by the author, when all other players submit numbers close to 64,000, the individual can increase his/her chance of winning by submitting a number around 42,667. However, if all other players reason the same way, they can be better off by changing their strategies and submitting a number close to 42,667, which triggers the individual to send an even smaller number. Because only one person can win, there will always be players who are unsatisfied with the result and have the incentive to switch strategies, which breaks the equilibrium.
On the other hand, the conundrum appears to be more complicated and multi-dimensional as it applies game theory into reality. Instead of having only several players with a manageable number of strategies (what we are used to solving in class), this game can have numerous players, each with thousands of strategies, which makes the computation an almost impossible task. Moreover, it would be naïve to assume that each player in this game makes his/her choice independently. In a real-life scenario, people can and will chat about the interesting game they participated in with those around them. Such conversations add additional variables to people’s decision-making process. Furthermore, players in this game don’t have to be rational, which challenges the theoretical basis of game theory. In the article, the author points out that people might not care about winning and just send in random numbers. All of the above contribute to the complexity of the problem and drive the game away from pure theoretical arguments. Therefore, it is evident that people’s behaviors in reality are hard to predict even with a sophisticated theory. From another perspective, the theory’s incapacity of prediction and lack of power in the real life is reassuring—our reality and people’s daily interactions are so vivid, diverse and creative that they can’t be simply concluded by a theory. I guess that’s why the study of people remains an absorbing and intriguing topic across time.
https://www.washingtonpost.com/business/kominerss-conundrums-pick-a-number-any-number-at-all/2021/09/12/28decf9e-13ca-11ec-a019-cb193b28aa73_story.html