Tuesday, May 8, 2012

Game Changer

Watch the video above before reading the blog. If you are not able to watch it here, you can watch it using the link: http://www.youtube.com/watch?v=Onwn9EmX1GI 

I saw this video about two years ago and shared it on my personal blog. On the face of it, the story seems quite evident: After barking obscenities at each other from across the gate for so long, the two brave-hearts want to keep it the same way, even when there is nothing to stop them. However, an analysis suggests that this behavior might have roots in game theory.
Let us first look at the payoffs when there is a gate between the two players. Barking at each other gives satisfaction, so let this across-the-gate pleasure have a payoff of 2. If no one barks the payoff is 0. Being able to bark alone will give maximum satisfaction so let that payoff be 3. Being barked at, without a quid pro-quo, is tantamount to humiliation, and therefore has a negative payoff of -1. Clearly there is a nash equilibrium here, and our friends seem to know it.
Now let us look at the game without the gate. Absence of the gate changes the game entirely. Barking now can lead to an actual fight which might have extreme negative consequences for each of the two players. Let this negative payoff be -20. Let the payoff of a bark-off without a fight be 5. (Notice that this payoff is more than across-the-gate pleasure of barking which is 2). The expected payoff of barking when the gate is not between the players, with equal probabilities of either event, is therefore 0.5 X -20 + 0.5 X 5 = -7.5. And of course, there is nothing like barking alone with no gate in between. So that payoff will be very high; let it be 6. Likewise, being barked at, with no response and with no gate to blame on is utter humiliation. So, let the payoff be -4 in this case.

Now comes the tricky part. What is the payoff for circling around each other? The fact that the two players don’t bark at each other when there is no gate in between suggests two possibilities:

  1. If the payoff is greater than 6 then there is a Nash equilibrium where each player is better off not barking at each other.
  2. If the payoff is less than 6 then there is no Nash equilibrium, and it is difficult to ascertain why our friends kept quiet on all the three occasions.
What is it that is making them behave in this way: Nash equilibrium, disposition, or severity of the negative payoff? Perhaps, that is what the two brave-hearts are trying to figure out when they move in circles around each other. The white one at one point seems confused whether it is the same ‘enemy’ or some parallel universe across the gate, but let us leave that dimension for now.

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