Analysis of Card Game with Bluffing

Analysis of Card Game with Bluffing

The card game with bluffing is analysed as follows.

First we recall the game mechanics:

There are two players: $A$ and $B$.

First it is $A$'s move.

$A$ receives one of two possible cards: either the high card or the low card.

If he receives the high card, he must bid $2$ credits.

If he receives the low card, he has two options, either:

$\map A 1: \quad$ He may pay $1$ credit, and the play is complete.

or

$\map A 2: \quad$ He may bid $2$ credits.

If $A$ has bid $2$ credits, it is $B$'s move.

$B$ has two options, either:

$\map B 1: \quad$ He may pay $1$ credit

or

$\map B 2: \quad$ He may challenge $A$'s bid.

If $A$ had the high card, $B$ must pay $2$ credits to $A$.

If $A$ had the low card, $A$ must pay $2$ credits to $B$.

The play is complete.

Strategies of $A$

If $A$ receives the high card, the bid is compulsory.

If $A$ receives the low card, the bid is optional.

Let $A_1$ be the strategy that $A$ pays $1$ credit.

Let $A_2$ be the strategy that $A$ bids $2$ credits.

Strategies of $B$

If $A$ pays, $B$ has no option but to receive $1$ credit.

If $A$ bids, then $B$ has $2$ strategies.

Let $B_1$ be the strategy that $B$ challenges $A$'s bid.

Let $B_2$ be the strategy that $B$ pays $1$ credit.

The outcome

There are $4$ possible combinations of strategies:

$\left({A_1, B_1}\right)$:

If $A$ had the low card, $A$ pays $1$ credit to $B$.

If $A$ had the high card, $B$ challenges $A$'s bid and so pays $2$ credits to $A$.

There is a $50 \%$ chance of either one happening.

The mean payoff to $A$ is therefore $\dfrac {2 - 1} 2 = \dfrac 1 2$ credit.

$\left({A_1, B_2}\right)$:

If $A$ had the low card, $A$ pays $1$ credit to $B$.

If $A$ had the high card, $B$ pays $1$ credit to $A$.

There is a $50 \%$ chance of either one happening.

The mean payoff to $A$ is therefore $\dfrac {1 - 1} 2 = 0$ credits.

$\left({A_2, B_1}\right)$:

If $A$ had the low card, $A$ bids $2$ credits, $B$ challenges $A$'s bid and wins, and so $A$ pays $2$ credits to $B$.

If $A$ had the high card, $A$ bids $2$ credits, $B$ challenges $A$'s bid and loses, and so $B$ pays $2$ credits to $A$.

There is a $50 \%$ chance of either one happening.

The mean payoff to $A$ is therefore $\dfrac {2 - 2} 2 = 0$ credits.

$\left({A_2, B_2}\right)$:

If $A$ had the low card, $A$ bids $2$ credits, $B$ accepts $A$'s bid and $B$ pays $1$ credit to $A$.

If $A$ had the high card, $A$ bids $2$ credits, $B$ accepts $A$'s bid and $B$ pays $1$ credit to $A$.

In both cases the payoff to $A$ is $1$ credit.

The mean payoff to $A$ is therefore $1$ credit.

$\blacksquare$

Sources

• 1956: Steven Vajda: The Theory of Games and Linear Programming ... (previous) ... (next): Chapter $\text{I}$: An Outline of the Theory of Games: $2$