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:

- If the
payoff is greater than 6 then there is a Nash equilibrium where each
player is better off not barking at each other.
- 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.

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