Example — Strega and Gigante

A two-player fantasy game: with one player’s mixed strategy fixed, we look for the other’s best response by computing their expected winnings as a function of the probability of play.

Example — Strega/Gigante vs Cavaliere/Drago

Player 1 chooses Strega (S) or Gigante (G); Player 2 chooses Cavaliere (C) or Drago (D). The plays are independent (both play blind). The payoff matrix $(v_1; v_2)$ is:

G1 / G2	Cavaliere	Drago
Strega	$(1; -2)$	$(-2; 3)$
Gigante	$(3; -1)$	$(-3; -3)$

Suppose G1 has already fixed the mixed strategy $P(S)=1/3$, $P(G)=2/3$. Find G2’s optimal strategy (parameter $p=P(C)$, $1-p=P(D)$).

Joint probabilities (independent plays): $P(S,C)=\tfrac{1}{3}p,\quad P(S,D)=\tfrac{1}{3}(1-p),\quad P(G,C)=\tfrac{2}{3}p,\quad P(G,D)=\tfrac{2}{3}(1-p).$

G2’s expected winnings (sum over the four profiles of the payoffs $v_2$ times their respective probabilities):

\bar V_2(p) &= (-2)\cdot\tfrac{1}{3}p + 3\cdot\tfrac{1}{3}(1-p) + (-1)\cdot\tfrac{2}{3}p + (-3)\cdot\tfrac{2}{3}(1-p) \\ &= -\tfrac{2}{3}p + 1 - p -\tfrac{2}{3}p - 2(1-p) \\ &= -\tfrac{7}{3}p + 1 - 2 + 2p \\ &= \tfrac{-7+6}{3}p - 1 \\ &= -1 - \tfrac{p}{3}. \end{align*}$$ **Conclusion (line decreasing in $p$).** $\bar V_2(p)=-1-p/3$ is *decreasing* in $p$: G2 minimises their loss by choosing $p=0$, that is *always Drago*. G2's optimal winnings are $\bar V_2(0)=-1$ (they always pay $1$ EUR on average).

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Topics: Probabilita
Concepts: Matrice di payoff · Strategia mista · Valore atteso
Methods: Strategia mista · Valore atteso
Skills: Calcolo probabilita · Modellizzare