The Good, The Bad, The Ugly & The Game Theory
Though weather is manageable, I caught a cold. Not worrying, but I might be needing a rest, at a moment I need it the least: Game Theory is getting stiff: The conditions for a strategy-proof monetary policy are trickier than assumed, but hey, I shall be posting on that very shortly.
I was just rewatching “The Good The Bad & The Ugly” by Sergio Leone. There are movies that one can never get enough of watching over and over again, and Leone’s tetralogy are no exception. I particularly like the last scene, where Tuco (Ellie Walach), Angel/Sentenza (Lee Van Cliff) and Blondie (Clint Eastwood) compete remorselessly for a treasure left by confederate troops.
The scene I am referring to is the final duel, when the three protagonists face each other, and set on killing for the confederate gold worth $ 200,000. Actually, I was listening to the movie while working on Osborne’s ‘Game Theory’ (I know, it is not much productive, but there are bad habits one enjoys thoroughly) I could not help myself but to link both things: isn’t the final scene a good illustration for game theory? Since many of my friends advised me to write lighter pieces, I shall use this one as an experiment: how to do some game theory with Sergio Leone (and your humble servant too)
There are three duellers: The Good (Blondie), The Bad (Sentenza) and the Ugly (Tuco). They compete for a certain amount of money ($ 200,000), hidden in a certain tomb, whose name’s Blondie (the Good) wrote on a piece of rock. In essence, all players are equal, due to the fact that they have the same amount of ammunition, and when they shoot, they cannot fire quickly enough and kill the two others! If they do draw, it’s against one and only one other dueller, and so, there is no possibility for collusion (as in: ‘Blondie, let’s get rid of this pig Angel!’ or ‘Tuco, you’re a good man, you should get some bucks out of Blondie’s corpse.’)
These conditions reduce the game to a basic form: which opponent each dueller is going to shoot? Payoffs are also pretty clear: the player either gets money or gets killed. Is there equilibrium here? In layman’s terms, is there a situation whereby duellers get the money and do not change their mind if they are offered to re-do the duel?
In fact, it seems there is. Not because the movie suggests so (as it later turns out, Blondie had Tuco’s pistol emptied the night before they reached the cemetery. In that sense, Blondie had a strictly dominant strategy of shooting Sentenza, which he did) but because the whole structure of the game incentive each dueller to implicitly coordinate with another to bring down the third: regardless of Blondie’s trick, both Blondie and Tuco fired –or tried to- on Angel. Why so, and was it equilibrium? If it was to be done again, would one of them at least change their mind?
Now, the game is imperfect: all three duellers have a history together. They know that of the three, Angel is the most ruthless and no one benefits from allying with him. On the other hand, Blondie and Tuco have a history of lucrative cooperation.
The table shows clearly that is it better to team up with with someone against the third dueller, rather than just draw and shoot at random (payoffs are sufficiently clear for them not to randomize. In gibberish economics, that means there’s already a Nash equilibrium, so randomizing is not necessary – the outcome of Nash equilibrium is obtained by mixed strategies). We need however, to introduce another condition: because all three protagonists dealt with each other before, they have preferences over their respective ‘partner’. It is reasonable to consider their preferences as follows:
For Tuco: U>G>B
For Blondie: G>U>B
For Sentenza: B>U>G
(note: duellers are considered selfish, so they prefer to take all the bounty to themselves, then choose another protagonist when needs to be) Because of that ordering, and even though pay-offs are the same when teaming-up occurs, both Blondie and Tuco have incentive to cooperate and shoot Sentenza, which eventually happened. In essence, there was little high-brow rationality, especially for Tuco: he is the exclusive second choice of both duellers, and he is better off with Blondie than he is with Sentenza -per his own preferences-.
Why does it constitute an equilibrium? Because it dominates all other moves, i.e. duellers -at least Blondie and Tuco- are better off and do not want to wish to change from their choices: Blondie prefers to split the money with Tuco, and so does the latter. Sentenza is know to be ruthless and has every incentive to keep it that way. It is thus a stable equilibrium.
Afrinomad put to me this proposal: can we find some Game Theory application to the current wave of revolutionary fervour in the MENA region? I must confess I got very enthusiastic over this idea: however intermediate my knowledge in game theory is, I’d very much like to give it a shot. So I would like to post on my own terms: ‘are the people of MENA ready to throw over their rulers, under Game Theory settings?’. Stay Tuned!