2011年5月16日

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Just Something About E-Card in Kaiji

Just watched Kaiji last night, which was a movie quite a long time ago and the E-Card at the last part of the movie had drawn my attention.
The E-Card is played by 2 players and it consists of 10 cards that are made up of 2 sets of cards, one set contains 1 Emperor, 4 Citizens and another one contains 1 Slave and 4 Citizens.

The rule is quite simple and straightforward, each player select a card and compare between them. The Emperor conquers a Citizen, it leads to a Draw if a Citizen meets a Citizen, and a Slave is defeated by a Citizen.

Emperor->Citizen->Slave->Emperor

where -> depicts the conquering sequence

Thus, I've done some calculation on the probability of winning and losing of the two players.
Note that the total combination of two individual cards is 5x5=25.

If both players play without flipping the card up, means they play completely blindly and randomly, the probabilities of winning and losing for the first round are as below:

SLAVE PLAYER
WIN: 1/25 [only 1 chance for the Slave to meet the Emperor]
LOSE: (1/5*4/5)+(4/5*1/5)=8/25 [probability of a Slave to meet a Citizen and a Citizen to meet Emperor]
DRAW: 4/5*4/5=16/25 [a Citizen meets a Citizen]

EMPEROR PLAYER
WIN: (4/5*1/5)+(1/5*4/5)=8/25 [probability of a Citizen to meet a Slave and an Emperor to conquer a Citizen]
LOSE: 1/25 [only 1 chance for the Emperor to meet the Slave]
DRAW: 4/5*4/5=16/25 [a Citizen meets a Citizen]

Obviously, the player holding the Slave card is disadvantaged if compared to the player holding Emperor Card since there is a probability of losing of 8/25 in the first round of game!

If the player selects a card before comparing, the probabilities of winning and losing for the game are as below:

SLAVE PLAYER
(SLAVE as the first card)
WIN: 1/5
LOSE: 4/5

(Citizen as the first card)
WIN: 0/5
LOSE: 1/5
DRAW: 4/5

EMPEROR PLAYER
(Emperor as the first card)
WIN: 4/5
LOSE: 1/5

(Citizen as the first card)
WIN: 1/5
LOSE: 0/5
DRAW: 4/5

So we can see that the winning/losing of the first round determines the whole game's result. So, normally both player won't use the Slave/Emperor Card in the first round. But if both players place 3 Citizen Cards consecutively, it will eventually lead to a 1/2 of chance to defeat or get defeated by the opponent.

This E-Card game is even more psychologically complex than playing the Poker Card!