```    1/*
2Truel, or duel among three opponents.
3There are three truelists, a, b and c, that take turns in shooting with a gun.
4The firing order is a, b and c. Each truelist can shoot at another truelist
5or at the sky (deliberate miss). The truelist have these probabilities of
6hitting the target (if they are not aiming at the sky): a 1/3, b 2/3 and c 1.
7The aim for each truelist is kill all the other truelists.
8The question is: what should a do to maximize his probability of living?
9Aim at b, c or the sky?
10Note that the best strategy for the other truelists and situations is
11easy to find intuitively and corresponds to aim at the best shooter.
12See https://en.wikipedia.org/wiki/Truel
13Martin Shubik, Game Theory and Related Approaches to Social Behavior, 1964, page 43
14
15*/
```

?- `best_strategy(a,[a,b,c],S)`. % What is the best action for a? % S= sky ?- `mc_sample(survives_action(a,[a,b,c],0,b),100,P)`. % What is the probability that a survives if it aims at b? % P = 50/189=0.26455026455 ?- `mc_sample(survives_action(a,[a,b,c],0,c),100,P)`. % What is the probability that a survives if it aims at c? % P = 59/189=0.31216931216 ?- `mc_sample(survives_action(a,[a,b,c],0,sky),100,P)`. % What is the probability that a survives if it aims at the sky? % P `25/63` 0.39682539682 ?- `mc_sample(survives([a,c],a,0),100,P)`. % What is the probability that a survives against c? % P =1/3 ?- `mc_sample(survives([a,b],a,0),100,P)`. % What is the probability that a survives against b? % P`3/7`0.42857142857 ?- `mc_sample(survives_round([b],[a,b],a,0),100,P)`. % What is the probability that a survives against b when it's b's turn? %P`1/7`0.14285714285 */

```   40:- use_module(library(mcintyre)).   41
42:- if(current_predicate(use_rendering/1)).   43:- use_rendering(c3).   44:- use_rendering(graphviz).   45:- endif.   46:- dynamic kr/1,num/1.   47:- mc.   48
49
best_strategy(+A:atom, +L:list, -S:atom)
The best strategy for truelist A with L still alive is to aim at S (with sky for the sky).

/

```   58best_strategy(A,L,S):-
59    delete(L,A,L1),
60    append(L1,[sky],L2),
61    maplist(ev_action(A,L,0),L2,LP),
62    sort(LP,LP1),
63    reverse(LP1,[_P-S|_]).```
ev_action(+A:atom, +L:list, +T:term, +S:atom, -C:couple)
Tuelist A with L still alive performing action S in turn T survives with probability P in C=P-S.

/

```   73ev_action(A,L,T,S,P-S):-
74  mc_sample(survives_action(A,L,T,S),100,P).```
survives_action(+A:atom, +L0:list, +T:term, +S:atom)
A survives truel performing action S with L0 still alive in turn T

/

```   82survives_action(A,L0,T,S):-
83  shoot(A,S,L0,T,L1),
84  remaining(L1,A,Rest),
85  survives_round(Rest,L1,A,T).```
shoot(+H:atom, +S:atom, +L0:list, +T:term, -L:list)
When H shoots at S in round T and L0 still alive, the truelist still alive become L /
```   93shoot(H,S,L0,T,L):-
94    (S=sky ->
95      L=L0
96    ;
97      (hit(T,H) ->
98        delete(L0,S,L)
99      ;
100        L=L0
101      )
102    ).
103
104
105hit(_,a):1/3.
106
107hit(_,b):2/3.
108
109hit(_,c):1.```
survives(+List:list, +Individual:atom, Round:term)
Individual survives the truel with List at Round

/

```  117survives([A],A,_):-!.
118
119survives(L,A,T):-
120    survives_round(L,L,A,T).```
survives_round(+Rest:list, +List:list, +Individual:atom, +Round:term)
Individual survives the truel at Round with Rest still to shoot and List truelist still alive /
```  128survives_round([],L,A,T):-
129  survives(L,A,s(T)).
130
131survives_round([H|_Rest],L0,A,T):-
132    base_best_strategy(H,L0,S),
133    shoot(H,S,L0,T,L1),
134    remaining(L1,H,Rest1),
135    member(A,L1),
136    survives_round(Rest1,L1,A,T).```
base_best_strategy(+A:atom, +T:list, -S:atom)
the best action for A when T is the list of surviving truelist, is S

These are the strategies that are easy to find (most intuitive)

/

```  148base_best_strategy(b,[b,c],c).
149base_best_strategy(c,[b,c],b).
150base_best_strategy(a,[a,c],c).
151base_best_strategy(c,[a,c],a).
152base_best_strategy(a,[a,b],b).
153base_best_strategy(b,[a,b],a).
154base_best_strategy(b,[a,b,c],c).
155base_best_strategy(c,[a,b,c],b).
156
157remaining([A|Rest],A,Rest):-!.
158
159remaining([_|Rest0],A,Rest):-
160  remaining(Rest0,A,Rest).
161