Triell

The trial is a variant of the duel with three instead of two fighters (each against each other, not two against one). The triell became known for its paradoxical property that under certain conditions good shooters are at a disadvantage compared to bad shooters.

Like a duel, a trial can be played in different ways. For example, it must be agreed whether only one shot may be fired or whether the shooting will take place until only one shooter survives.

Trielle have been scientifically researched in various forms, in particular by D. Marc Kilgour. A full investigation in terms of game theory is still pending.

history

The Triell first became known in 1959 when it was mentioned in Martin Gardner's book “Mathematische Rätsel undproblem”. As the earliest source he gives "Question Time" by Hubert Phillips (Farrar and Rinehart, Inc., 1938). However, he found the solution in a scientific publication, the American Mathematical Monthly, December 1948 issue.

The idea soon became very popular with the well-known western "Il buono, il brutto, il cattivo" (1966, German title: " Two glorious scoundrels ") by Sergio Leone . For the final argument with revolvers over a pot of gold, three Westerners line up in a circle. Each of the three has different motives for shooting at one of the others. Leone uses the complex interweaving of motifs to multiply the tension of a classic duel.

In 1973, after further scientific research , Donald Knuth found an optimal strategy that Gardner and his predecessors had overlooked: All players shoot in the air.

In the 1970s, the Triell was scientifically investigated using a wide variety of rules:

Trielle with an infinite number of shots and a limited number of shots per player.
Trielle, in which aerial shots were forbidden or allowed.
Triels in which the order of the firing right is varied.
Triels in which duels between two surviving players continue under completely new conditions, etc.

The triell became popular once more when it was mentioned in Simon Singh's book “Fermat's Last Sentence” (2000). Compared to Gardner's version, the hit probabilities have been changed there. However, Donald Knuth's result was not taken into account in the solution.

In the short film “Triell” (2004, director: Su Turhan ) with Michael Ballhaus in front of the camera, three men fight in a triell for a woman, played by Bettina Zimmermann .

analysis

Gardner and Singh believed that the weakest shooter would shoot best in the air for their own benefit, as would the top two shooters at each other. However, this is only applicable under certain conditions that were not mentioned by either author.

In the following, the basic features of Kilgour's solution ( Lit .: Kilgour, 1975) are set out for the triell as it is described in the versions by Gardner and Singh .

The problem

Three shooters organize a triell in which they shoot one after the other until only one participant is still alive. All three shooters are known to be of different skill. To make the triell fair, the worst is given the first shot; then comes the second best (if he's still alive), then the best (if he's still alive). Then it starts again with the worst. Where should the shooters ideally aim?

Additional assumptions

For a clear determination of the optimal strategies, some properties of the combatants' utility functions must be clearly defined.

The utility of shooter 2 when shooters 1 and 3 are shot is u ₂(13) .

Then the following assumptions are plausible: ${\ displaystyle u_ {2} (13) \ geq u_ {2} (1) = u_ {2} (3)> u_ {2} (123) \ geq u_ {2} (12) = u_ {2} ( 23) \ geq u_ {2} (2)}$

With further assumptions on the utility function, the optimal strategies can now be determined.

The solution

Scenario 1. "non-hostile" players

Are all players "non-hostile", i. H. if they don't care how many other players are dead, as long as they stay alive themselves (that would mean for the utility function of the second player :) , then the only solution of the game is shooting all players in the air. ${\ displaystyle u (123) = u_ {2} (12) = u_ {2} (23) = u_ {2} (2)}$

Scenario 2. At least one “enemy” player

However, if one of the players is "hostile" (e.g. ), then there is exactly one of the following solutions - depending on the probability of the individual players hit: ${\ displaystyle u_ {2} (23)> u_ {2} (2)}$

If G ₂ > 0: All shoot at each other.
If G ₂ <0: The two strongest shoot each other. The weakest shoots in the air.
If G ₂ = 0: The two strongest shoot each other. The weakest will shoot the strongest with any probability.

It is ${\ displaystyle G_ {2} (a, b, c) = - 1 + b + b ^ {2} c + a \ left (b + cb ^ {2} -2bc-b ^ {2} c \ right) + a ^ {2} b ^ {2} c \; \; \; f {\ ddot {u}} r \; \; \; 0 <a \ leq b \ leq c <1}$

Example: For a = 1/3, b = 2/3 and c = 1, G ₂ ( a , b , c ) = -0.24926, i.e. H. the weakest would shoot in the air and the other two at each other.

The following table shows the great advantage that the weakest shooter has in the trial. In the first example (Gardner), there is a 54% chance that he will survive. The best shooter, who always hits, has only a 10% chance of winning (both statements for the scenario of at least one “enemy” player ).

Table: Optimal strategies and survival probabilities in two exemplary triales

fighter	TW	optimal strategy	ÜW analytical	ÜW analytical	ÜW simulated	TW	optimal strategy	ÜW analytical	ÜW simulated	optimal strategy	ÜW
	At least one "enemy" player									"non-hostile" players
	by Martin Gardner					by Simon Singh
A.	1	Shot at B	24.167%	9/90 (10.00%)	10.44%	1	Shot at B	14/63 (22.22%)	22.75%	Shot in the air	1.00
B.	0.8	Shot at A	31.111%	32/90 (35.56%)	35.32%	2/3	Shot at A	24/63 (38.10%)	38.00%	Shot in the air	1.00
C.	0.5	Shot in the air	44.722%	49/90 (54.44%)	54.24%	1/3	Shot in the air	25/63 (39.68%)	39.25%	Shot in the air	1.00

TW stands for the hit probability of the respective shooter. It was simulated 10,000 times each time. The event of survival or not is binomially distributed for every shooter. The mean survival probability in the simulation has a spread that is proportional to if the number of attempts is. This explains the discrepancies between the simulation results and Gardner's analytical results from the third digit ( ). ${\ displaystyle {\ sqrt {1 / n}}}$ ${\ displaystyle n}$ ${\ displaystyle {\ sqrt {1/10000}} = 10 ^ {- 2}}$

If the survival probabilities are used as a benefit for the players, you are automatically in the scenario of "non-hostile" players .

The difficulty in analyzing the triell is not to calculate the survival probabilities of the players if the optimal strategies of the players are already known, but to find out the optimal strategies first.

Criticism of the solution

Some points of criticism can be found in the above analysis. Only stationary strategies are permitted, which means that no strategy change can be made during the exchange of fire. In addition, the influence of risk attitudes ( e.g. risk aversion ) is not explicitly taken into account.

literature

Martin Gardner: Mathematical Puzzles and Problems. Vieweg Braunschweig 1968
D. Marc Kilgour: The Sequential Truel. International Journal of Game Theory 4, 3, 1975, 151-174.
D. Marc Kilgour: Equilibrium Points of Infinite Sequential Truels. International Journal of Game Theory 6, 3, 1978, 167-80.

Web links

Wiktionary: Triell - explanations of meanings, word origins, synonyms, translations

The Truel List - An English-language list of publications, websites and films that deal with triells or contain triall-like situations.
The worst shooter survives . From: Praxis der Mathematik 59 (2014), pp. 45–47
Triell - the short film
The Good, the Bad and the Ugly in the German-language IMDB