A truel is the three-player extension of a duel, i.e. a three-player game where the objective of each player is to eliminate the other two. Each player has an intrinsic shooting ability, or marksmanship, that determines the probability of hitting the target when they fire. The game, which was originally proposed by Kinnaird (1946), has multiple versions depending on e.g. whether the players shoot simultaneously or sequentially (and, if sequentially, whether the shooting order is predetermined, or determined at random from among the survivors), the number of rounds, the number of attempts given to each player to eliminate the others, or whether deliberate misses are allowed (Kilgour 1971; Kilgour 1975; Kilgour 1977).

As Prof. Hernández often points out in his lectures, one of the most interesting features of this game is that truels can reproduce the somewhat paradoxical –yet empirically observed– fact that mediocrity triumphs in many social contexts: in general, the most skilful truelist is not the one with the highest chance of winning the contest, and indeed there are many conditions for which it is actually the least skilled of the three truelists who has the highest chance of surviving the truel (see an example in Colman (1995, pp. 273-274)).

In this appendix we only consider sequential truels. In the *random*
sequential truel, one of the truelists is chosen at random. The
selected truelist has the option to choose which other player to aim
at; then, with a certain probability dependent on his marksmanship, he
will eliminate the targeted opponent. Regardless of the result, the
process is repeated until there is only one survivor. In the *fixed*
sequential truel, there is a fixed order for the elimination attempts
which is determined by the truelists' marksmanship. The least
skilled truelist is given the first opportunity to eliminate another
player; then the next truelist in skill makes his attempt (assuming he
is still alive); and then, it is the most skilful truelist's
turn. This sequence is repeated until only one truelist remains.

To analyse the game, let us call the most skilful truelist
"player *A*", the next in skill
"player *B*", and the least
skilled truelist "player *C*".
Let us also make the usual game theoretical assumptions (i.e. common
knowledge of skills and rationality). Under these conditions, there are
two strategies that, depending on the nature of the truel and the
players' marksmanships, may appear as Nash equilibria
(Amengual and Toral 2006):

- Strategy
*BAA*, in which player*A*initially tries to eliminate player*B*, and players*B*and*C*initially attempt to eliminate player*A*. This strategy is known as*strongest-opponent strategy*. - Strategy
*BA0*, in which player*C*deliberately misses his chance to eliminate*A*and*B*until either of them has been eliminated by the other.

The rationale to discuss this game here is twofold. Firstly, the truel game has been recently analysed using the same framework of stochastic processes and Markov chains that we explain in the paper (see Toral and Amengual 2005). Toral and Amengual (2005) analyse different versions of the sequential game, and analyse them as Markov chains with three absorbing states (i.e. the possible victories of each one of the players). The second reason is that Amengual and Toral (2006) also present an interesting analysis of a variation of the truel game where physical space plays a role. This more sophisticated model, which is named "collective spatial truel", is commented below.

The "collective spatial truel" introduces new assumptions in the model. Three different groups of players characterised by their marksmanship are considered. The players are randomly located on a bi-dimensional lattice (20×20 cells) in some fixed proportions. The neighbourhood of a player is defined as the Von Neumann neighbourhood of radius 1. A new fundamental assumption included in this version is that any two players belonging to the same group (and hence with the same marksmanship) will not try to eliminate each other. The rest of the rules are as follows:

- One of the remaining players in the lattice is chosen randomly.
- The selected player chooses two other players from his neighbourhood randomly and plays a truel with them. If there is only one neighbour, they play a duel. If there is no neighbour, the player moves to a randomly chosen neighbouring site. In any case, players belonging to the same group do not attempt to eliminate each other, so it is possible that more than one player survives the truel (or the duel). Every player uses the strongest-opponent strategy.
- Steps 1 and 2 are repeated until there is only one surviving group in the lattice.

This model can be seen within the Markov chain framework. For
that, we can define the state of the system as a vector of dimension
400 representing each cell of the lattice. Every component of this
vector can take four different values, three to indicate a player of
each of the groups and a fourth value to denote that the cell is empty.
Thus, the number of possible states is 4^{400}.

Deriving the whole transition matrix of a system of such dimension seems rather futile. Having said that, note that most values in the transition matrix are zero, and it is not difficult to calculate any given transition probability. The only possible moves from any given state correspond to the occurrence of a truel, a duel, or a random move to a neighbouring site. All these possibilities imply a change in 2 values of the 400-dimensional state vector at the most. Thus, the transition probability between two states that differ in more than 2 values is necessarily zero. Furthermore, the positions of the 2 values that may change at the most must correspond to cells that belong to the same neighbourhood, and at least one of these two cells must change from being occupied to being unoccupied.

The exact probability of any possible transition could be
derived from the probability of selecting each surviving player, from
the probabilities of choosing different combinations of duels and
truels given the selected player's neighbourhood, and from
the analytical results derived by Toral and Amengual (2005) to
calculate the probability of winning a truel for each player, in both
random and fixed sequential truels. Nonetheless, as the authors
themselves point out *"… because the
analytical treatment seems rather difficult, let's look at
the results from a direct numerical simulation of the aforementioned
rules …"* (Amengual and Toral 2006, pp. 93).

The important point for our purposes is to realise that this model has many absorbing states. (A state is absorbing if and only if all the players in the grid belong to the same group.) Thus, applying proposition 3 we can state that the model is not ergodic, i.e. the probability of ending up in any one particular absorbing state depends on the initial conditions. This is the reason for which the authors consider different initialisations of the model to calculate the type of predominant absorbing state in the model.

AMENGUAL P and Toral R (2006) Truels, or Survival of the
Weakest. *Computing in Science and Engineering*,
8(5), pp. 88-95

COLMAN A M (1995) *Game Theory and its Applications
in the
Social and Biological Sciences*. Oxford, UK.:
Butterworth-Heinemann.

KILGOUR D M (1971) The simultaneous truel. *International
Journal of Game Theory*, 1(1), pp. 229-242

KILGOUR D M (1975) The Sequential Truel. *International
Journal
of Game Theory*, 4(3), pp. 151-174

KILGOUR D M (1977) Equilibrium points of Infinite Sequential
Truels. *International Journal of Game Theory*, 6(3),
pp. 167-180

KINNAIRD C (1946) *Encyclopedia of Puzzles and
Pastimes*.
Secaucus, NJ: Citadel.

TORAL R and Amengual P (2005) Distribution of winners in truel
games. In Garrido P L, Marro J, and Muñoz M A (Eds.)
*Modelling Cooperative Behavior in the Social Sciences*.
Vol. 779 of AIP
Conf. Proc.: 128-141. American Institute of Physics