Five-Card Partial Results

For those who thrive on jumbles of numbers, here are the payoff matrices for the five-card game after either: The bids are annotated with the probability (shown in parentheses) that each player should assign to choosing that bid. E.g., after winning the 5 with 5 vs 1, if the next card is the 4, the player who lost the 5 should bid 3 or 5 with equal probability, and the player who won the 5 should bid 1 or 4 with equal probability.

The net payoff for each matrix is shown to the right of the matrix. In each case it is in terms of the player who won the first card (the player whose bids are represented by selecting a column in the matrix).

"5" won, 5 vs 1

Interestingly, which player comes out ahead in this situation depends on the next card. If it's the 4, the game is still even. If it's 3 or 1, the player who bid 1 in the first round is slightly ahead. If it's 2, the player who bid 5 is slightly ahead. Averaging across the four cases, however, the player who bid 5 is behind, with an expected payoff of about -0.073.
5(5,1)
4		Bid & Weight
1
(1 / 2)		2
(0)		3
(0)		4
(1 / 2)
Bid
&
Weight		2
(0)		-1+79/135+1+1
3
(1 / 2)		-1-1+79/135+1
4
(0)		-19/105-7/30-1+79/135
5
(1 / 2)		+1+4/35-11/15-1
=0
5(5,1)
3		Bid & Weight
1
(44695 / 89896)		2
(0)		3
(0)		4
(45201 / 89896)
Bid
&
Weight		2
(0)		-1+118/1159+1+1
3
(0)		-1-1+118/1159+1
4
(40565 / 44948)		-4/35-47/180-1+118/1159
5
(4383 / 44948)		+1+1-1/3-1
=
-253
-----
44948
5(5,1)
2		Bid & Weight
1
(774 / 33681)		2
(0)		3
(27545 / 33681)		4
(5362 / 33681)
Bid
&
Weight		2
(5800 / 33681)		-1-1/21+2/15-1/2
3
(11669 / 33681)		-1/3-4/9-1/21+1/3
4
(16212 / 33681)		+11/18+8/450-1/21
5
(0)		+1+1+13/60-1
=
+653
------
101043
5(5,1)
1		Bid & Weight
1
(0)		2
(3597 / 10492)		3
(6895 / 10492)		4
(0)
Bid
&
Weight		2
(6895 / 10492)		-4/9+1/291-4/35-1
3
(3597 / 10492)		0-2/9+1/291+4/35
4
(0)		+1+2/90+1/291
5
(0)		+1+1+19/42-4/9
=
-75237
-------
1017724

"1" won, 2 vs 1

In the solutions to these matrices the fractions start to get rather long, so the weights and expected payoffs are given as decimals. For the calculation of these numbers I wish to thank Geoffrey Gerdes for his web page interface for solving linear programming problems.
1(2,1)
5		Bid & Weight
1
(0.26786)		3
(0.0691)		4
(0)		5
(0.66304)
Bid
&
Weight		2
(0.05962)		-1+1+1+334/975
3
(0.22843)		-1-16/117+1+71/154
4
(0)		-19/42-1-16/117+1
5
(0.71195)		+4/90-1-16/117
=
+0.0283768
1(2,1)
4		Bid & Weight
1
(0.06132)		3
(0)		4
(0.32999)		5
(0.60870)
Bid
&
Weight		2
(0)		-1+1+64/99+1/2
3
(0.11706)		-1+2/9+2/3+1/3
4
(0.56856)		+13/45-1+2/9+4/9
5
(0.31438)		+1+1/2+1/2+2/9
=
+0.3615756
1(2,1)
3		Bid & Weight
1
(0.38001)		3
(0.42398)		4
(0.19600)		5
(0)
Bid
&
Weight		2
(0.28117)		-1+1+3/10-1
3
(0.28583)		+1/3-1393/4845+1/2+1/3
4
(0.43300)		+2/3-2/9-1393/4845+1/3
5
(0)		+1+1+1/18-1393/4845
=
+0.1027710
1(2,1)
2		Bid & Weight
1
(0.29566)		3
(0.18009)		4
(0.52424)		5
(0)
Bid
&
Weight		2
(0.40413)		-5581/10989+1/2+1/2-1
3
(0.24684)		+46/195-46/195+1/3-7/30
4
(0.34903)		+1+1/6-46/195+1/6
5
(0)		+1+1+1/6-46/195
=
+0.2020103