The Good, the Bad, the Ugly, and Game Theory

"Il Buono, il Brutto, il Cattivo" ("The Good, the Bad and the Ugly"), a masterpiece directed by Sergio Leone and with the performances of Clint Eastwood, Lee Van Cleef and Eli Wallach, is according to Tarantino “The best-directed film of all time”. Two elements stand out from the point of view of decision-making theory, the first is that the leitmotif of the cooperation between “Blondie” (Eastwood) and “Tuco” (Wallach) is the information asymmetry about the exact place where the treasure is located; while the second is its final scene, where an unprecedented confrontation occurs until that moment in the western genre, a triple duel ("truel", to use an expression coined by M. Shubik) between the three protagonists that reaches its climax amid the wonderful music composed by Ennio Morricone.

Typically, in the western genre, duels involve time-related decisions, where "quick draw," or the ability to rapidly draw a pistol and fire it accurately on a target in the process, is an essential skill for survival. However, the inclusion of more gunfighters opens another strategic dimension, since you must decide who to shoot. An example where both dimensions intersect is when a lone gunman faces off against several, as in the memorable scene in "Unforgiven" - one of the three films in the genre that have won the Academy Award for Best Picture - in which Beauchamp tells Munny that when facing a group, "an experienced gunman always shoots the best shooter in the group first" and Munny replies that he “was lucky with order, but I've always had it when it comes to killing”.

The first time I read about mathematical models of truel was in Martin Shubik's book "Game Theory in the Social Sciences" (1982), and it made such an impression on me that I dedicated a section of my graduate thesis to discuss the subject. The first mention of this tripping situation is in C. Kinnaird's "Encyclopaedia of puzzles and pastimes" (1946). Two relatively recent works that make a formal exposition of the subject are the paper by Kilgour and Brams ("The Truel”, Mathematics Magazine, vol. 70, no. 5, 1997) and the one of Toral and Amengual ("Distribution of Winners in Truel Games”, Modelling Cooperative Behaviour in the Social Sciences, vol. 779, 2005).

For the study of tripping from the perspective of game theory, certain assumptions must be made that make the rules of the game explicit. First, we assume that a player's aim is not dependent on his opponent. Second, Truelists fire sequentially, one after another, if one Truelist misses a shot, then it is another Truelist's turn. Third, each truelist has the right to choose which person to target. Finally, the ammunition is infinite, so that there can only be one survivor and asymptotic reasoning can be applied.

In a random truel, this algorithm 1 is used. One of the remaining truelists is chosen at random. 2. The chosen truelist selects an opponent and, with a certain probability (marksmanship) eliminates that opponent from the game. 3. Whatever the outcome, steps 1 and 2 are repeated until only one survivor remains, the winner. We are going to call the truleists “A”, “B” and “C”, and for convenience, we will denote the probability of hitting (marksmanship) as a, b, and c, and the probability of survival as P_A, P_B, P_C.

These rules characterize the so-called “random sequential truel”, and it should be clear that assuming simultaneous shots, fixed order, or finite ammunition changes the nature of the game significantly. In this type of game, two issues have given it relevance in terms of practical applications (i) In many cases, the best shooter is not the one with the highest probability of survival, (ii) the pacts are unstable.

To understand these results, it should be noted that the rational choice of each player, if they want to maximize their probability of survival, is to shoot the best shooter within the options they have. But it is good to go progressively since it is clear that if it were a duel (for example, between A and B, and assuming a> b) random sequential, it can be shown that the best shooter would have the highest survival probability (P_A=a/(a+b)>b/(a+b)=P_B). But when we come to consider truels, the choice of whom to shoot is the strategic choice, so we can say that A's strategies form the set {B, C}, and so on. A strategic profile would be an element of the Cartesian product of the strategic profiles of each player and would be an ordered triplet, where the first element is A's strategy and so on. Each strategic profile can be associated with a vector of the type (P_A, P_B, P_C) since the survival probability of each shooter depends on their own choices and those of others. Now we can prove the first paradoxical result, and for this, we will use table No. 1, which assumes a = 1, b = 0.8 and c = 0.5.

Let us now consider the CCB profile, where A and B shoots C, and C shoots B. It is easy to see that C, for example, can improve his survival probability if he changes his strategy and chooses to shoot A (it goes from 0.072 to 0.085 ). So now consider the CCA profile, and B could now improve his prospects if he changes his strategy and shoots A, and in that case, C has no incentive to change his strategy (it even improves his survival probability). Finally, by reviewing the CAA profile, it can be verified that A can improve his survival probability if he chooses to shoot B. The BAA profile has an interesting property since it is not possible for any player to improve his survival probability by changing strategy, so this profile constitutes a "Nash equilibrium" for this game, and in fact, it is the only one there is.

The paradox appears here, since A being a shooter who never misses, his survival probability is the lowest, while C being a mediocre shooter has the highest survival probability. Maintaining the assumption that A always hits, there are many marksmanship combinations where A will have the best chance of survival, but B and C will be required to be too erratic. On the other hand, B will have the advantage only if it is almost as good as A and C remains mediocre. However, if we assume that B also has 100% successes, A can't obtain the maximum probability of survival, and C will obtain it if his aim exceeds 62%.

The second point requires that shooters be allowed to communicate before the truel. In a first example, A can approach C and threaten to shoot him if he decides to attack him and misses, and if C believes in that threat he could move the outcome of the game to BAB, where the probability of A increases and that of B falls. The problem is that A's threat is not credible, since, if C ignores it, A must move to a CAA situation where its probability of survival is reduced.

Another case of interest is that A and B agree to dispatch C and both remain in a duel, which would give both a higher probability of survival (CCA). Again, the promise is incredible, and to verify it let's think that it is A's turn and he knows that he can get rid of B without being able to punish him for violating the pact. If it is B's turn, he may think that A is going to betray him and that it is better to shoot him at once.

The Father of Ethology, Konrad Lorenz, commented to Martin Shubik that it is not uncommon for three bitter rivals living close to each other to attack each other. Jack London tells in White Fang that three wolves: (i) "the boss”, a strong young adult, (ii) "the one-eyed wolf”, a cunning old wolf, and (iii) an ambitious three-year-old cub, disputed the right to be the mate of a formidable she-wolf. Had they followed the rational strategy of attacking the strongest first, the result would have tended to favour the cub, but the two strongest wolves decided to be irrational and agreed to get rid of the cub before engaging in a decisive duel.