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:
 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
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 playerStu or Pollydoes the game favor?
Source: H. Nahikian, Topics in Modern Mathematics, 1966, pp. 232233
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:
 Set up the 3x3 payoff matrix (w.r.t. Stu's view, his three plays are Ace of Diamonds, Ace of Clubs, and 2 of Diamonds...with corresponding payoffs 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 minimax 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
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?
