Optimal strategy for Rock, Paper, Scissors

Contents

Classic version of Rock, Paper, Scissors

Players C and R play rock, paper, scissors:

After each round the loser pays the winner 1$. This is is shown in the payoff matrix A: player C picks a column, player R picks a row

$$ A = \left[ \matrix{0 & 1 & -1 \cr -1 & 0 & 1 \cr 1 & -1 & 0} \right] $$

Here the optimal strategy for each player is

Then expected winnings for player C = expected loss for player R are zero, i.e., the game is fair.

A = [0 1 -1; -1 0 1; 1 -1 0]
S = 2;
Am = A + S;   % add some S such that all entries of Am are positive
              % answer p,q,cost does not depend on choice of S
b = [1;1;1];
c = [1;1;1];

[x,costm,y] = linearoptim(Am,b,c)
p = x/sum(x)
q = y/sum(y)

costC = 1/sum(x) - S
costR = 1/sum(y) - S

cost = q'*A*p % expected winnings for player C = expected loss for player R

A =
     0     1    -1
    -1     0     1
     1    -1     0
x =
      0.16667
      0.16667
      0.16667
costm =
          0.5
y =
      0.16667
      0.16667
      0.16667
p =
      0.33333
      0.33333
      0.33333
q =
      0.33333
      0.33333
      0.33333
costC =
     0
costR =
     0
cost =
   -6.163e-33

Rock, Paper, Scissors with modified payoffs

Now we play Rock, Paper, Scissors with a modified payoff matrix A:

Now the optimal strategy is different:

Then expected winnings for player C = expected loss for player R are -.15686. Therefore the game is not fair, it is advantageous for player R.

A = [0 1 -2; -3 0 4; 5 -6 0]
S = 10;
Am = A + S;   % add some S such that all entries of Am are positive
              % answer p,q,cost does not depend on choice of S

[x,costm,y] = linearoptim(Am,b,c)
p = x/sum(x)
q = y/sum(y)

costC = 1/sum(x) - S
costR = 1/sum(y) - S

cost = q'*A*p

A =
     0     1    -2
    -3     0     4
     5    -6     0
x =
     0.039841
     0.035857
     0.025896
costm =
      0.10159
y =
     0.061753
     0.026892
     0.012948
p =
      0.39216
      0.35294
       0.2549
q =
      0.60784
      0.26471
      0.12745
costC =
     -0.15686
costR =
     -0.15686
cost =
     -0.15686


Published with MATLAB® R2018a