In a rock-paper-scissors variant, a win with paper scores 5, a win with rock scores 1 and a win with scissors scores 2, with the loser scoring the negative of that. What mixed strategy should a player use, and what is the value of the game?

Step-by-step solution

1. 1

   Write down the payoff matrix. Let the row player choose rock, paper or scissors with probabilities $p$, $q$, $r$, where $p+q+r=1$ and all are non-negative. Paper beats rock ($+5$), rock beats scissors ($+1$), scissors beats paper ($+2$), and each loss is the negative. The game is zero-sum and symmetric — both players face the same matrix — which already tells us its value must be $0$.

2. 2

   Compute the row player's expected payoff against each pure reply.

   $E_{\text{rock}}=0\cdot p+5q-1\cdot r=5q-r,$
   $E_{\text{paper}}=-5p+0\cdot q+2r=-5p+2r,$
   $E_{\text{scissors}}=1\cdot p-2q+0\cdot r=p-2q.$

3. 3

   Impose the equalising condition. At a mixed equilibrium the opponent must be indifferent between all three replies, otherwise they would switch to the best one and exploit us. So

   $5q-r=-5p+2r=p-2q=v,$

   together with $p+q+r=1$. This "make the opponent indifferent" principle is the entire method.

4. 4

   Solve the linear system. From $5q-r=p-2q$ we get $p=7q-r$. Substituting into $p+q+r=1$:

   $(7q-r)+q+r=8q=1\;\Longrightarrow\;q=\tfrac18.$

   From $5q-r=-5p+2r$ we get $5p+5q=3r$; with $p=\tfrac78-r$ this becomes $\tfrac{35}{8}-5r+\tfrac58=3r$, so $8r=5$ and $r=\tfrac58$, hence $p=\tfrac78-\tfrac58=\tfrac14.$

5. 5

   Read off the value and verify. Substituting into any of the three expressions, $v=p-2q=\tfrac14-\tfrac14=0$. Checking all three: $5q-r=\tfrac58-\tfrac58=0$ ✓, $-5p+2r=-\tfrac54+\tfrac54=0$ ✓, $p-2q=0$ ✓, and $p+q+r=\tfrac28+\tfrac18+\tfrac58=1$ ✓. The value $0$ confirms the symmetry argument from step 1.

6. 6

   Interpret the weights. Scissors is played most often ($\tfrac58$) even though winning with it scores only $2$ — because scissors is the counter to paper, the opponent's most rewarding weapon, and it must be discouraged. Paper, the highest-scoring winner, is played least ($\tfrac18$). In equalising games the probability of a move tracks how badly the opponent's best option must be suppressed, not how much that move itself pays.