Refining Nash Equilibrium: Should a Mother Confiscate Her Son’s Phone?
The linked article talks about the brief history of John Nash and his Nash equilibrium. The first part touches on the classic prisoner’s dilemma. Here is the story: two prisoners face the same deal offered by the attorney. If they both confess, they will be sentenced then years in jail. If one stays quiet while the other snitches, then the snitch will be freed while the other will be sentenced to life. But if both of them keep quiet, then they will only face a one year sentence. John Nash pointed out that the solution to this problem is when both prisoners confess to the attorney, and this solution is the well-known Nash Equilibrium. Nash Equilibrium, counter-intuitively, does not necessarily describe a solution that maximizes the total payoff of everybody. Instead, it considers the decisions made by each person independently. To put it simply, in a Nash Equilibrium, nobody can have better payoff by switching to other strategies. The fact that the total welfare isn’t actually maximized naturally lead to the second bullet point in the article: Crowd Trouble.
Crowd Trouble was well illustrated by the traffic control example in the lecture. Assuming 4000 passengers commute from A to B, the travel time is indicated on the route below. Travel time on routes AC and DB is dependent on the number of passengers travelling (indicated as x and y). An equilibrium solution is when the two routes ACB and ACD takes the same time, that is, when x = 2000 and y = 2000. The total travel time for each passenger is therefore 65. In other words, half of the passengers will go through ACB while the other half goes through ADB.
An interesting phenomenon happens when a route from C to D with zero travel time is added to the traffic plan. All passengers will choose the route ACDB as a result. And the total travel time becomes 85, which is even more than the time when there is no connection between CD. To understand why this is possible, we can put ourselves in the shoes of a passenger: when everyone is taking the route ACDB, we won’t be able to make our travel time better by switching to any other route. The new route act like a vortex, which sucks in all the passengers from the other routes. This is what we call Nash Equilibrium.
The most interesting part of article is about refining the Nash Equilibrium. Refinement mostly centers upon incorporating real life situations into the formulation. For example, one refinement involves accounting properly for non-credible threats. It gives the example of a teenager who threatens to run away from home if his mother confiscate his mobile phone. In this case, the mother’s cost is the need to worry about his son wandering around outside. In other words, her payoff is not to make herself worry if she gives her son the mobile phone. Naturally the Nash Equilibrium is that she gives her son the phone. However, Reinhard Selten argues that the mother does not have to worry much about it because to her son, a night out on the street is worse than not having a mobile phone, and therefore the threat is not well supported. The best outcome is hence the mother confiscating her son’s mobile phone and making him do his homework. Mr. Selten’s work is important because it reduced the number of Nash Equilibria, addressing real-life situations in a better way.
Link: http://www.economist.com/news/economics-brief/21705308-fifth-our-series-seminal-economic-ideas-looks-nash-equilibrium-prison