Analysis of VNM POKER(4,4,3,5)

Prerequisites: All Theory Chapters, except maybe Chapter 9, and the first part on poker.

This page demonstrates the difficulties one faces when looking for mixed Nash equilibria in larger examples, and also that, although one player's Nash equilibrium mix draws against much more strategies than the other player's Nash equilibrium mix, playing such a mix may still be worthwhile in two-player zero-sum games.

• 1. Mixed Nash equilibria
• 2. Performance of pure strategies against the mixed Nash equilibria
• References
• Exercises and Projects

	FFFF	FFFC	FFCF	FFCC	FCFF	FCFC	FCCF	FCCC	CFFF	CFFC	CFCF	CFCC	CCFF	CCFC	CCCF	CCCC
CCCC	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
CCCR	0.15	0.00	0.28	0.13	0.28	0.13	0.42	0.27	0.28	0.13	0.42	0.27	0.42	0.27	0.55	0.40
CCRC	0.55	0.02	0.40	-0.13	0.68	0.15	0.53	0.00	0.68	0.15	0.53	0.00	0.82	0.28	0.67	0.13
CCRR	0.70	0.02	0.68	0.00	0.97	0.28	0.95	0.27	0.97	0.28	0.95	0.27	1.23	0.55	1.22	0.53
CRCC	0.95	0.42	0.42	-0.12	0.80	0.27	0.27	-0.27	1.08	0.55	0.55	0.02	0.93	0.40	0.40	-0.13
CRCR	1.10	0.42	0.70	0.02	1.08	0.40	0.68	0.00	1.37	0.68	0.97	0.28	1.35	0.67	0.95	0.27
CRRC	1.50	0.43	0.82	-0.25	1.48	0.42	0.80	-0.27	1.77	0.70	1.08	0.02	1.75	0.68	1.07	0.00
CRRR	1.65	0.43	1.10	-0.12	1.77	0.55	1.22	0.00	2.05	0.83	1.50	0.28	2.17	0.95	1.62	0.40
RCCC	1.35	0.82	0.82	0.28	0.82	0.28	0.28	-0.25	1.20	0.67	0.67	0.13	0.67	0.13	0.13	-0.40
RCCR	1.50	0.82	1.10	0.42	1.10	0.42	0.70	0.02	1.48	0.80	1.08	0.40	1.08	0.40	0.68	0.00
RCRC	1.90	0.83	1.22	0.15	1.50	0.43	0.82	-0.25	1.88	0.82	1.20	0.13	1.48	0.42	0.80	-0.27
RCRR	2.05	0.83	1.50	0.28	1.78	0.57	1.23	0.02	2.17	0.95	1.62	0.40	1.90	0.68	1.35	0.13
RRCC	2.30	1.23	1.23	0.17	1.62	0.55	0.55	-0.52	2.28	1.22	1.22	0.15	1.60	0.53	0.53	-0.53
RRCR	2.45	1.23	1.52	0.30	1.90	0.68	0.97	-0.25	2.57	1.35	1.63	0.42	2.02	0.80	1.08	-0.13
RRRC	2.85	1.25	1.63	0.03	2.30	0.70	1.08	-0.52	2.97	1.37	1.75	0.15	2.42	0.82	1.20	-0.40
RRRR	3.00	1.25	1.92	0.17	2.58	0.83	1.50	-0.25	3.25	1.50	2.17	0.42	2.83	1.08	1.75	0.00

Obviously this example is more complex than the case S=2 discussed here. An analysis of this family of games is no longer possible for general parameters r, m, and n, so we choose concrete values r=4, m=2, n=3. Even this concrete case is difficult enough to analyze.

Remember that pure strategies for Ann are four-letter words of "C"s and "R"s, and Beth's pure strategies are four-letter words of "F"s and "C"s. The full normal form is calculated using the Excel sheet VNMPoker4.xls. Moreover, we showed in the previous chapter that some pure strategies are weakly dominated. After eliminating them we got

Applying Brown's fictitious play, running say 1000 rounds in the Excel file BROWN10.xls, it is fairly convincing that Ann should choose 3/4 of strategy CCCR and 1/4 of RCCR. Beth's result is less clear, according to the results I obtained, Beth should mix about 5% of FFFC, about 37% of FFCC, and about 58% of FCCC. Fortunately, all we need to know is that Ann mixes only CCCR and RCCR in order to find the exact solution. Concentrating on these two of Ann's pure strategy, we get the following matrix:

Now let us use the graphical method for 2 × n zero-sum games. We draw four straight lines: The FFFC line from (0,0) to (1,49/60), the FFCC line from (0, 2/15) to (1,5/12), and so on. The height of the FFFC line at position x indicates Ann's payoff if Ann plays a mix of x of RCCR and 1-x of CCCR, and if Beth plays FFFC, and similar for the other three lines. Since Beth wants to maximize her own payoff, and therefore wants to minimize Ann's payoff, for each such x Beth would respond by the pure strategy that has the lowest height for this x-value. According to the graph, for 0 ≤ x ≤ 1/4 (a lot of CCCR), Beth would play FFFC, for 1/4 ≤ x ≤ 0.89, Beth would play FCCC, and otherwise CCCC. The curve on the top of the gray area indicates the payoff Ann can achieve when playing a mix of x of RCCR and 1-x of CCCR. Since Ann wants to maximize her payoff, she chooses that x where this curve has the largest height, so she chooses x=1/4. The expected payoff for Ann is 49/240 ≈ 0.2.

Note that it is a little odd, a coincidence, that three straight lines meet there, FFFC, FFCC, and FCCC. That means that Beth would mix these three strategies. To find the possible percentages q₁, q₂, q₃=1-q₁-q₂, we need again the Indifference Theorem and a little Algebra.

We know that both CCCR and RCCR are Ann's best responses to Beth's mix of q₁ of FFFC, q₂ of FFCC, and 1-q₁-q₂ of FCCC. Ann's payoff in both cases are on the left and the right of the following equation:

Distributing the expressions, and at the same time raising all fractions to obtain the common denominator 60, we get

Since none of the probabilities can be negative, q₂ must be between 0 and 15/32 = 0.46875, q₁ must be between 0 and 15/64 = 0.234375, and q₃ must be between 17/32 = 0.53125 and 49/64 = 0.765625.

The relationship between q₂ and q₁ is linear, a linear function, as well as the relationship between q₃ and q₁. So the graphs of these functions are straight lines. In the chart below, q₂, q₃, and q₁ are displayed as functions of q₁. Every vertical line indicates a Nash equilibrium mix. The one shown, for instance, refers to q₁ = 6%, q₂ = 34%, and q₃ = 59%.

Playing a Nash equilibrium strategy in a zero-sum game guarantees a certain expected payoff, the value of the game, against any one of the other player's strategies. But the sad part is that this Nash equilibrium will not have a higher expectation against many other strategies of the other player, which are different to the Nash equilibrium counterpart. So, against sophisticated play, some less sophisticated play is not punished. In VNM POKER(2,r,m,n) for instance, all a player needs to know when playing against the Nash equilibrium is to avoid to fold or to check with the higher value card. What about VNM POKER(4,4,3,5)?

Let us now look how pure strategies of Beth perform against Ann's optimal mix of 3/4 CCCR and 1/4 RCCR: Of course we need the values in the full normal form in VNMPoker4.xls for the calculations, see tab "some mixes".

• FFFC, FFCC, FCFC, and FCCC are optimal against Ann's Nash mix. If Beth mixes these four pure strategies in any way, she will expect to lose only 49/240 ≈ 0.2
• CFFC, CFCC, CCFC, CCCC have an expectation of about -0.3 for Beth, when played against Ann's Nash mix.
• FFFF, FFCF, FCFF, FCCF have an expectation of about -0.49 against Ann's Nash mix.
• CFFF, CFCF, CCFF, and CCCF are Beth's weakest pure strategies, since they yield an expectation of about -0.58 for Beth.

The following chart, based on the tab "some mixes" in VNMPoker4.xls, shows how Ann's pure strategies perform against Beth's Nash equilibria mixes of q₁ of FFFC, 15/32 - 2·q₁ of FFCC, and 17/32 + q₁ of FCCC, for q₁ varying from 0 to 15/64 = 0.234375. Note that we don't include FCFC here, since this would further complicate matters. The graph displays Ann's payoff versus q₁.

The highest point of the gray area is just in the middle of the graph, thus Ann mixes 50% of CCRR and 50% of RCRR. As above, we can conclude that Beth mixes 15/32 of FFCC and 17/32 of FCCC.

So is this another Nash equilibrium? No, since Ann's best response to Beth's mixing 15/32 of FFCC and 17/32 of FCCC are again only CCCR and RCCR. So our assumption of a Nash equilibrium for Ann mixing CCRR and RCRR results in a contradiction, and is therefore not possible.

From the above discussion we obtain that there are infinitely many Nash equilibria, Ann mixing 75% of CCCR and 25% of RCCR, and Beth mixing q₁ of FFFC, 15/32 - 2·q₁ of FFCC and FCFC, and 17/32 + q₁ of FCCC, for 0 ≤ q₁ ≤ 15/68 ≈ 0.2206. For large q₁ the percentage for FCFC is probably limited further. Ann's payoff in each of these Nash equilibria equals 49/240 ≈ 0.2. Since we made assumptions on what pairs of strategies are used in Ann's Nash mix, we cannot exclude the possibility for further Nash equilibria.

There is another conclusion from the graph above: Since most of Ann's pure strategy payoffs against the mixes are smaller for small q₁, a clever Beth might choose q₁ = 0 to profit more from Ann making mistakes and not mixing CCCR and RCCR against Beth's mix. So Beth might just mix 15/32 of FFCC and 17/32 of FCCC. This Nash equilibrium might be considered to be a little more reasonable than the other Nash equilibria.

From these results there follows that if Ann plays the Nash mix mentioned, she will not gain an additional advantage if Beth plays any mix of the four pure strategies FFFC, FFCC, FCFC, and FCCC. This translates into a behavior strategy of "F??C". You should always fold with a lowest value, and always call with a highest value, but otherwise you can do whatever you want. But every mixed strategy that sometimes calls with a lowest value and sometimes folds with a highest value will perform sub-optimally against Ann's optimal mix. On the other hand, a Beth playing any Nash equilibrium mix (with q₁ strictly smaller than 15/68) will not gain an advantage against an Ann using any mix between CCCR and RCCR. Such a mix translates into the behavior strategy "?CCR" of always checking with a value of 2 or 3, always raising with a highest value of 4, but doing whatever with a lowest value of 1. However, every mixed strategy that sometimes raises with a value of 2 or 3, or sometimes checks with a value of 4, will perform sub-optimally against Beth's optimal mix of FFCC and FCCC. Decide yourself whether you would have chosen any of these sub-optimal mixes. All opponents except Ali Baba, Jim Knopf, U Hu, and I Can in the applet AppletVNMPoker4.html, for m=3 and n=5, play sub-optimal, either as Ann or Beth. Jim Knopf plays just the Nash equilibria mixes discussed. Jim Knopf, U Hu, and I Can play Nash equilibria mixes for m=3, n=4, respectively m=1, n=2, respectively m=1, n=3. You can check the long-term performance of these strategies, and some more provided by students, in the applet AppletVNMPoker4CC.html.

• John von Neumann, Oskar Morgenstern, Theory of games and economic behavior, Princeton University Press 1953, 330.182 V89
• Gambit ...

Exercises

1. Assume you are playing many rounds of VNM POKER(2,r,m,n) as Beth against a first moving opponent Ann who assumes that you always call with a card of value 2, and who tries to estimate your ratio of FC versus CC as described above, and then play the best response. You certainly can mislead Ann by occasionally folding with a card of value 2, but the real question is whether such play could increase your expected value. Find out.