# Definition:Card Game with Bluffing

## Contents

## Definition

The **card game with bluffing** is a class of games which is exemplified by the example whose mechanics are as follows:

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:

or

- $A (2): \quad$ He may bid $2$ credits.

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

$B$ has two options, either:

- $B (1): \quad$ He may pay $1$ credit

or

- $B (2): \quad$ 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.

### Payoff Table

The payoff table of the **card game with bluffing** is as follows:

$\text B$ | ||

$\text A$ | $\begin{array} {r {{|}} c {{|}} c {{|}} } & B_1 & B_2 \\ \hline A_1 & 1/2 & 0 \\ \hline A_2 & 0 & -1 \\ \hline \end{array}$ |

## Analysis

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:

or

- $A (2): \quad$ He may bid $2$ credits.

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

$B$ has two options, either:

- $B (1): \quad$ He may pay $1$ credit

or

- $B (2): \quad$ 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 $1$ credits to $A$.

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

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

$\blacksquare$

## Solution

From the payoff table:

$\text B$ | ||

$\text A$ | $\begin{array} {r {{|}} c {{|}} c {{|}} } & B_1 & B_2 \\ \hline A_1 & 1/2 & 0 \\ \hline A_2 & 0 & -1 \\ \hline \end{array}$ |

The solution is:

- $A$ takes strategy $A_1$ for $2/3$ of the time, and $A_2$ for $1/3$ of the time.
- $B$ takes strategy $B_1$ for $2/3$ of the time, and $B_2$ for $1/3$ of the time.

## Sources

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