Zero-Sum Poker Signalling Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Zero-Sum Poker Signalling Game

• Players: Player 1, Player 2, and Nature.
• Strategies: Nature draws a card for each player, $H$ or $L$; Player 1 observes own card and chooses Pass or Raise; after a raise, Player 2 chooses Fold or Stay.
• Rules: Each card is high or low with equal probability; cards are shown at the end; the higher card wins; win payoff is $8$, loss payoff is $-8$, and a tie gives $0$.

Payoff Details

• The four card states are

• Since each player receives $H$ or $L$ with probability $\frac{1}{2}$, each state occurs with probability

• If Player 1 passes, cards are shown and the higher card wins:
  • $(H,H)\Rightarrow (0,0)$
  • $(H,L)\Rightarrow (8,-8)$
  • $(L,H)\Rightarrow (-8,8)$
  • $(L,L)\Rightarrow (0,0)$
• If Player 1 raises and Player 2 folds, the extracted tree gives

in every state

• If Player 1 raises and Player 2 stays, the extracted tree gives:
  • $(H,H)\Rightarrow (0,0)$
  • $(H,L)\Rightarrow (12,-12)$
  • $(L,H)\Rightarrow (-12,12)$
  • $(L,L)\Rightarrow (0,0)$
• The game is zero-sum in every terminal node.

Game Tree

Payoff Matrix

• Not used: the extracted material is an extensive-form poker tree rather than a compact normal-form matrix.

Derivation (Best Response Analysis)

Player 2 after a raise

• In state $(H,H)$, folding gives $-8$ and staying gives $0$, so Player 2 prefers Stay.
• In state $(H,L)$, folding gives $-8$ and staying gives $-12$, so Player 2 prefers Fold.
• In state $(L,H)$, folding gives $-8$ and staying gives $12$, so Player 2 prefers Stay.
• In state $(L,L)$, folding gives $-8$ and staying gives $0$, so Player 2 prefers Stay.

Player 1 before the response

• With card pair $(H,L)$, raising can improve Player 1’s payoff from $8$ to $12$ if Player 2 stays.
• With card pair $(L,H)$, raising can improve Player 1’s payoff from $-8$ to $8$ if Player 2 folds, but can also reduce it to $-12$ if Player 2 stays.
• In tied states $(H,H)$ and $(L,L)$, passing and a raise followed by stay both give $0$, while a fold after the raise gives Player 1 payoff $8$.
• Hence the strategic tension is bluffing by Player 1 versus calling by Player 2.

Zero-sum structure

• Every terminal payoff is of the form

• So one player’s gain is exactly the other player’s loss.

Nash Equilibrium

Result:

The extracted notes specify the full extensive-form game tree and terminal payoffs, but they do not provide a solved equilibrium strategy profile.

Social Optimum

• Not used: this is a zero-sum game, so there is no separate cooperative benchmark distinct from the strategic payoffs.

Insights

Insight:

• The game is a simple poker-style signalling problem: Player 1 acts after seeing a card, and Player 2 reacts to the raise.
• Raising with a low card creates a bluffing incentive because Player 2 may fold.
• Staying against a raise is valuable only when Player 2’s card is strong enough.
• Zero-sum structure makes the game suitable for mixed-strategy analysis.