# Zero Sum Games

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Zero Sum Games

Question One

A modified version of this game can be expressed as:
First Child
rock
paper
scissors
rock
0, 0
-1, 1
2, -1
Second Child

paper
1, -1
0, 0
-1, 1
scissors
-1, 2
1, -1
0, 0
qt represents the probability that Player 2 chooses the option of rock on a given round. qy represents the probability that
the player goes for paper, and qd the probability that the player opts for scissors. Because these are the only available options, then the probabilities have to satisfy qr+qp+qs=1.
The supposed value of the game for the first player if he/she applies a mixed strategy is:
If the first player takes his turn, the expected value is
Rock
0*qt + (-1)*qy+2*qd = 2qd-qy
Paper
1*qt+0*qy+(-1)*qd = qt-qd
scissors
(-1)*qt + 1*qy + 0*qd = qy-qt
From (1) and (2), we get
3qs = qt+qy .. (4)
(1) and (3), gives us

2qs = 2qy-qr .. (5)
(4) and (5) gives us

5qp = 3qd, implies qp = (5/3)qd
Substituting this in (4), we get qr = (12/5)qd
Using these values in qt+qy+qd=1 i.e.
(12/5)qd + qd+ (5/3)qd = 1
Simplifying this begets;

Qs = (1/4)
Therefore, qp = (5/3)*(1/4) = 5/12 and qr = (12/5)*(1/4) = (3/5)
Because the game is symmetric, player 2 also applies the same strategy.
Rock is chosen less in each round because the higher payoff connected to strategy rock, each player
would consider that his counterpart would choose rock. In this case, each player is entitled to a
payoff equal to zero. Therefore, each player would look to play paper or scissors.

Question Two

A zero sum game for two people involves a pay-off and a negative pay-off function for both players. The strategic form in this situation is expressed as (X, Y, A). In this case,

X represents a set of strategies for the first player

Y represents a nonempty set, the second player

A represents a real-valued function defined by the product of X × Y

Simultaneously, the two players make their choices unaware of what each has chosen.

In this kind of situation, neither of the players have a superior strategy because a chance of getting a pay-off depends on the counterpart’s decision. If a player has knowledge of his opponent’s choice, it makes it easy to exploit this and gain positive pay-off. Players will therefore use their instinct to guess what their opponent chooses. This represents the key process of randomizing a player’s behavior between H and T mixed strategies.

There are a number of ways to playing mixed strategy games depending on the game conditions. In the case of the pavement, the players randomize with a view of interpreting the thoughts of the opponent. As stated above, it is not possible to have clear knowledge of the opponent’s thoughts. Therefore, applying a random strategy presents an effective method of the interpreting the next move of the opponent.  For the first few attempts therefore, the results may be uncertain as each player makes new decisions. This implies that positive or negative pay-off cannot be determined. As the players start to understand the mindset of each other, choices are more likely to become similar. Therefore, this shows that payoff from the mixed equilibrium is only half as good for either player as either of the two pure equilbria.

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