All is Fair...Except Possibly in Card Games or...
Stu Dent and Polly Dent are playing a new game. Each player starts with an Ace of Diamonds and an Ace of Clubs...while Stu also has a 2 of Diamonds and Polly also has a 2 of Clubs.
The rules of the game:
Your Task: Is this a fair game? That is, over an extended time, will both players scores be approximately the same? If it is not a fair game, which player--Stu or Polly--does the game favor?
- On each play, each player selects a card to lay face down simultaneously, without seeing what card the other player selects
- The players then turn over their cards
- Stu wins if the suits match, while Polly wins if the suits do not match
- The winning player receives the numerical points shown on only his/her card (where an Ace's value is 1)
- But, if two 2's are shown, neither player receives any points
Source: H. Nahikian, Topics in Modern Mathematics, 1966, pp. 232-233
Hint: Play the game many (many!) times with a friend, keeping track of the accumulated score.
From each player's perspective, does there seem to be a particular card that should never be played?
Now, how can you explore the game mathematically?
Solution Commentary: This game was originally posed by H.W. Kuhn (Lectures on the Theory of Games, 1957), but was explored by H. Nahikian. In his solution, Nahikian suggested these steps:
Thus, thought the game seems fair, it is not fair...Stu is favored with an average win of 1/5 point per game. Does this fit your data?
- Set up the 3x3 pay-off matrix (w.r.t. Stu's view, his three plays are Ace of Diamonds, Ace of Clubs, and 2 of Diamonds...with corresponding pay-offs respectively of [1 -1 -2], [-1 1 1], and [2 -1 0].
- Stu eliminates Row 1 (e.g. [1 -1 -2]) as it is dominated by Row 3 (e.g. [2 -1 0])
- For the same reason, Polly next eliminates Column 3 (e.g. [1 0]) because it is dominated by Column 2 (e.g. [1 -1])...remember, columns, not rows!
- What remains is a 2x2 matrix with row values [-1 1] and [2 -1]...meaning Stu must choose from Ace of Clubs or 2 of Diamonds, while Polly must choose from Ace of Diamonds or Ace of Clubs
- Through easily accessible game theory techniques, the mini-max value of the game is calculated to be 1/5 based on the optimal strategies of Stu (0, 2/5, 3/5) and Polly (2/5, 3/5, 0)...that is, this means that Stu randomly plays the Ace of Clubs 2/5 of the time and the 2 of Diamonds 3/5 of the time