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An interesting card game

Here is a very interesting card game which has been introduced by Prof. Kandori at University of Tokyo.

There are two players, one is Red and one is Black. Each players have four cards in corresponding colors: King (K), Ace (A), 2 and 3.

Each round, both two players simultaneously (i.e, together) select one card from their hand, and determine the winner as the rule in the picture:
- If two cards are two Kings, Red wins.
- If one card is King and other card is not, Black wins.
- Otherwise, if number of two cards are different, Red wins. If number of two cards are identical, Black wins.

At first sight, it is very easy to say that the game is favor on Red.

But it is not true.

Even if both players play randomly, the Black has a bigger chance to win. You can check the simulation at:

It is easy to calculate the chance to win of Black is 9/16.

But we can go further.

Take into account the fact that K is very important card in this game, we can analyze:
- For Red, if he picks K, the chance to win is 1/4, and if he does not pick K, the chance is 1/2
- For Black, if he picks K, the chance to win is 3/4, and if not, the chance is 1/2.

Giving this analyzation, Red should pick K with the probability 1/3, and Black should pick K with the probability 3/5.

We simulated this game at:

Calculate exactly how much the chance to win of Red and Black may interest the readers.


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