Definition:Card Game with Bluffing

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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:

$\map A 1: \quad$ 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.


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:

$\map A 1: \quad$ 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.

$\blacksquare$


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.


This article contains statements that are justified by handwavery.
In particular: It is not made clear in the source work why this is. You just get: "This is indeed a solution, because no player can do better if the other sticks to his strategy, but could gain or lose more than $1/3$ (which is what $A$ gains now) if he departed from the optimal solution and his opponent took advantage of it." This is handwaving. A source work with a higher level of precision is needed here.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold.
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Sources

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