Winton (2003) decribes a simple coin toss game a follows:

"A fair coin is tossed, and the result is shown to player one. Player one must then decide
whether to pass or play. If player one passes, then he must pay player two $1. If player
one plays, then player two (who does not know the result of the coin toss) may either pass or play.
If player two pass, then he pays player one $1. If player two plays and the coin comes up heads,
then he pays player one $2. If player two plays and the coin comes up tails, then player one
must pay him $2."  
 

The game is further illustrated below:  

The objective is to find the optimal strategy for player one.  

Player one has four different alternatives available to him as seen below.  

We can also see the expected value for each alternative ie

Expected Value = Prob(Head)*Outcome_Head + Prob(Tail)*Outcome_Tail 

Since player1 selects first, he is never going to select Action1 and Action4 since his expected value

is negative if he selects these two strategies. The only attractive actions for player 1 are either

Action2 or Action4. Action2 is the most attractive sine it will maximize player1's expected return.

If we assume that player1 and player2 selections are completely random we can show this as follows:  

We should now note that Player1's expected return depends on the probability that Player1 selects

Action2 or Action4 and on the probability that Player2 selects pass or play
 

Player1's expected returns are again given by: 

We now assume that : 

Player1 expected value as a function of x1 and x2 is therefore given by: 

(1)

We can plot such expected value in a 3Dplot as follows: 

We can see that we have a saddle path equilibrium. This means that there exit one path where

the change in expected return for Player1 is constant. For example if we assume x1=0.1 then

we can see that player1's expected return is an increasing function of x2. If we assume x1=0.9 then

we can see that player1's expected return is an decreasing function of x2.   
 

If we assume x1=0.66, which is represented by the red line, then we can see that

player1's expected return is independent of x2.
 

We can show this as follows: 

Player 1's expected value when player2 picks pass is given by: 

(2)

Player 1' expected value when player2 picks play is given by: 

(3)

We can plot such expected values as follows: 

The two lines crosses at: 

	(4)

This means that player 1's optimal strategy is to pursue a mixed-strategy because if  

player1 selects such a strategy then player1 expected value is independent of x2.  

Player1 optimal strategy is therefore to select Action4 67 % of the time and Action2 33 %  

given that player2 selects pass with probability x2. Player2 has two pure strategies either pass

or play. We can show that the above strategy is optimal though simulation as follows: 

We can now assume that player2 knows the value of x1 ie x1=x2. 

This means that when when x1 is large ie when there is a large probability that player1

will select Action4 then the probability that player2 will select pass ie x2 is large and

when x1 is small ie when there is a large probability that player1 will select Action2

then the probability that player2 will select pass ie x2 is small. 

We can do some simulation and show that when both players pursues such Actions the game is

completely random. 

We assume that player1 also have the option to bluff: 

There are two ways that player1 can bluff and gain an advantage: 

1.A)  Player1 select Action2; Play on head and Play on Tail 

1.B) When he gets tail he pass instead of betting; Player1=-1  and Player2=+1 

1.D) If Player1 would not have bluffed the outcome would have been;  Player1=-2  and Player2=+2 

1.E) Which means that Player1 has gained +1 by bluffing. 

2.A)  Player1 select Action4; Play on head and Pass on Tail 

2.B) When he gets tail he bets instead of passing 

2.C) Player2 thinks player1 sticks to Action4 so player2 pass;  Player1=+1  and Player2=-1 

2.D) If Player1 would not have bluffed the outcome would have been;  Player1=-1  and Player2=+1 

2.E) Which means that Player1 has gained +2 by bluffing. 

The expected values are given by: 

Player1 expected value where x1=probability player1 selects Action4 and

x2=probability that player2 selects pass is therefore given by: 

(5)

We can plot such expected value as follows: 

We can see that the optimal strategy for player1 is to select Action4 50% of the time. 

The reason for this is that the uncertainty will be maximized at such point so player2

have a hard time determining if player1 is bluffing or not. 

The End !!