The Nash equilibria in a two-card poker game

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The Nash equilibria in a two-card poker game

Bachelor’s thesis: Economics and Business year 3 Supervisor: David Smerdon

Faculty: University of Amsterdam

Capacity group: General economics, Game theory Name: T.S.K. Tan Student number: 10574778 Date: 1st of February 2016 Place: Amsterdam Abstract

Poker is a rapidly growing gambling game, generating a revenue of €4,3 billion in 2015 and still increasing. It is a game of imperfect information, where multiple competing players must deal with probabilistic knowledge. Correct decision-making is essential to achieving high performance in a game of poker.

This thesis tries to find the existing Nash equilibria in a two-card poker game using the well-known poker models of Borel and Von Neumann as a

guideline. For a risk neutral poker player, a risk averse and a risk-seeking player the existing Nash equilibria are calculated. The results are highly divergent for the different player types.

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Page

Introduction 3

Terminology 4

1. The Borel and Von Neumann poker models 5

2. Nash equilibria in a two-card Borel poker game 7

3. Nash equilibria in a two-card Von Neumann poker game 16

4. A bridge from Borel to Von Neumann 21

5. Conclusion 27

6. Review 28

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Introduction

One of the most popular gambling games these days is poker. Due to the development of the Internet the global interactive poker gross winnings increased from €1,3 billion in 2004 to €4,3 billion in 2015 (retrieved from

http://www.statista.com/statistics/208468/global-interactive-poker-gross-win-since-2004/ on 23rd_{of January 2016). Many suggest that poker indeed is a}

gambling game. The Dutch government considers poker as a gambling game and taxes poker winnings with 29%, the tax on gambling.

On the other hand however, many people earn a relatively consistent living playing poker. To use a quote from the Hollywood movie “Rounders” - “Why do you think the same five guys make it to the final table of the World Series of poker EVERY YEAR? What, are they the luckiest guys in Las Vegas?” This quote alone obviously does not prove whether or not poker should be

categorized as a gambling game or not. Game theory does provide a more robust framework for exploring this query.

Not only is poker interesting to analyze from a juristic perspective, it is also interesting from a scientific point of view. Apart from luck, factors such as psychology, bluffing, strategy and acting with incomplete information come into play. To gain a better insight of the game and its components some simplified game theoretic models with just two players will be used in this thesis. Even though a regular poker game is played with more players, these models provide a good theoretical insight of the game.

Much research has been published on the variousstrategies in poker. One of the first and most famous poker models is that of the French

Mathematician, Émile Borel. Before WWII, in 1938, he published a poker model called “La Relance”. Another mathematician, John Von Neumann, has also done research on game theory with respect to poker. Together with Oskar Morgenstern he published his book Theory of Games and Economic

Behavior in 1944, in which a poker model with zero-sum games is introduced

(Ferguson, 2003). Bellman and Blackwell (1949), Bellmann (1952), and Karlin and Restrepo (1957) have further extended the work of Borel.

This thesis analyzes the strategy of a poker player within the context of Borel’s and Von Neumann’s poker models. The main goal is to gain a better understanding of the decisions made in a poker game and to find out whether an optimal strategy exists. In the first part the Borel and Von Neumann poker models are briefly introduced. Next, a simplified game of poker will be

played consisting of just two players and two possible cards. For the three possible player types, risk neutral, risk averse and risk seeking, the Nash equilibria will be calculated. In the final part of the thesis a third model is introduced, which can be seen as a bridge between the Borel and Von Neumann models. The research question linked to this thesis is: “What are the Nash equilibria in a two-card poker game using the Borel and Von Neumann poker models as a guideline?”

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Terminology

Ante: A small amount of money/chips that all the players are required to

make before they are given any card.

Bet: the first time a player can put money in the pot after having seen his

card(s).

Bluff: A bet done with a bad hand hoping the opponent(s) fold(s) to win the

pot.

Call: Contribute to the minimum amount of chips in the pot necessary to

continue playing a hand.

Check: Basically doing nothing, the next player then can decide what to do. Fold: Throwing away your hand/cards. You lose the possibility to win the pot

and your earlier bet.

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1. The Borel and Von Neumann poker models

Borel’s poker model: La relance

As mentioned in the introduction, Borel introduced a poker model called “La relance” in his book “Traite du Calcul des Probabilit́es et ses Applications Volume IV”.

There are several assumptions that are made in this model, which are as follows.This game of poker is played with just two risk-neutral players, each given an independent uniform hand on the interval [0,1]. Both players have two actions to pick from. Player 1 can “bet” or “fold” and player 2 can either “call” or “fold”. Before the players are dealt any cards they have to contribute an ante of 1 unit each. Both players are aware of their own cards, but not of their opponents’. Thus, the game that is being played here is a game of

imperfect information. The player with the highest card wins and receives the money in the pot. It has to be noted that the possibility of a tie is zero.

Strategy profile for both players: Player 1: X --> {bet, fold}

Player 2: Y --> {call, fold}

Let’s say that player 1 is given hand X and player 2 is given hand Y, where both X and Y have a uniform distribution on the interval [0,1]. Player 1 is first to move. If he decides to fold player 2 wins the pot without having to play any action. The utilities are therefore (utility player 1, utility player 2)=(-1,1). Player 1 can also bet an amount B, where B > 0. Now player 2 has to decide whether to play ‘call’ or ‘fold’. If player 2 chooses ‘call’ he has to put an extra amount B in the pot after which the hands of player 1 and 2 will be compared. The player with the highest card wins the pot. The utilities are as follows: (B + 1, -B - 1) if X > Y and (-B – 1, B + 1) if X < Y. If player 2 however chooses ‘fold’ instead of ‘call’ he loses and the utilities will be (1, -1).

Player 1 Bet Fold (-1, 1) Call Fold (B + 1, -B – 1) if X>Y (1, - 1) (-B – 1, B + 1) if X<Y

Figure 1 Game tree of the Borel poker model

In figure 1 the betting tree for the “La relance” is shown. In this tree the

winnings/utilities of players 1 and 2 are displayed after every possible move. Since the game that is being played here is a game with incomplete

information, the players do not know the hand of their opponent. There are however a few things that player 2 is aware of. If player 1 folds, player 2 knows that player 1 has a bad hand, as player 1 will not fold if he has a good

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hand. If player 1 chooses to play bet, player 2 does not know whether player 1 has a good hand or if he is bluffing.

Von Neumann model

The Von Neumann model is slightly different from the Borel model. Where it was possible to ‘fold’ in the Borel model, player 1 can now choose to play ‘check’. After playing ‘check’ the cards of both players will be compared. What this means is that if player 1 checks, player 1 has the chance to win the pot if his card is higher than that of player 2. This model is more favourable for player 1 since he does not immediately lose his ante if he does not want to bet.

Strategy profile for both players: Player 1: X --> {bet, check}

Player 2: Y --> {call, fold}

The game tree for the Von Neumann model is shown in figure 2. Player 1

Bet Check

(1, -1) if X>Y

Call Fold (-1, 1) if X<Y

(B + 1, -B – 1) if X>Y (1, - 1) (-B – 1, B + 1) if X<Y

Figure 2 Game tree of a Von Neumann poker model

These two models are quite a good representation of the truth since the hand each player can receive is continuous. However, this thesis will focus on a simplified game of poker where the deck of cards consists of just two cards, a high- and a low one. The number of high and low cards can be considered infinite, which implies that the chance of a high card remains 50% when the other player receives a low card.

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2. Nash equilibria in a two-card Borel poker game

Consider a one-shot two-card poker game with only a high card and a low card. The chance of getting a high card is 50% and so is the chance of getting a low card. There now is the possibility of both players getting a high- or low card at the same time since the deck is infinite. Furthermore, the assumption is made that both players start with a utility of 2 (the ante + B, which are both 1). The rest of the assumptions remain unchanged. To find out what the optimal strategy for each player is a game tree of the possible outcomes is constructed. Dealer High ( € 1 2) Low ( € 1 2) 1 1 Fold Fold Bet Bet (1, 3) (1, 3) - - - Dealer - - - High ( € 1 2) Low ( € 1 2) High ( € 1 2) Low ( € 1 2) 2 2 2 2

Call Fold Call Fold Call Fold Call Fold (2, 2) (3, 1) (4, 0) (3, 1) (0, 4) (3, 1) (2, 2) (3, 1)

Figure 3 Game tree of a two-card Borel poker game with an infinite deck

In order to find the possible Nash equilibria it is important to take into

account the players’ expectations. As mentioned before, the chances of a high and a low card are both 50%. It is however not rational for player 2 to think that the chance of player 1 having a high card is still 50% if player 1 bets. It will more likely be higher than 50%.

The assumptions each player has about the other player are as follows: Player 1 has the following assumptions about player 2:

• If player 2 gets H, player 2 plays ‘call’ with probability β1 (and ‘fold’

with 1 – β1)

• If player 2 gets L, player 2 plays ‘call’ with probability β2

Player 2 has the following assumptions about player 1:

• If player 1 gets H, player 1 plays ‘bet’ with probability α1

• If player 1 gets L, player 1 plays ‘bet’ with probability α2

For now consider that β1 > β2 and α1 > α2.

After having determined the assumptions that both players have about one another the expected utility for each action can be computed. There are three types of players with three corresponding expected utility functions.

Risk neutral: EU = E(x) Risk averse: EU = E(√x) Risk seeking: EU = E(x2

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Since the deck of cards consists just of two types of cards, high or low, both players can have four strategies. Player 1 can play ‘bet’ no matter what card he has, he can ‘bet’ with a high card and ‘fold’ with a low card, he can play ‘fold’ with a high card and ‘bet’ with a low card and he can decide to ‘fold’ any card. The same strategies are available for player 2 except player 2 can play ‘call’ instead of ‘bet’. However, for player 2 it depends on player 1 whether he is allowed to make an action, since the game could end before player 2 can decide what to do.

It may not be a surprise that both players want to maximize their expected utility. To kick off we shall look at what strategies yield the highest expected utility for two risk neutral players. The expected utility for both players consists of three parts:

- The utility if the other player plays bet/call with H, times the subjective probability that the other player plays bet/call.

- The utility if the other player plays bet/call with L, times the subjective probability that the other player plays bet/call.

- The utility if the other player plays folds, times the subjective probability that the other player folds.

The calculations for player 1’s expected utility for each action is: EU(bet|H) = € 2 ∗β1 2 + 4 ∗ β2 2 + 3 1 − β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3 − β1 2 + β₂ 2 EU(bet|L) = € 0 ∗β1 2 + 2 β2 2 + 3 1 − β1 2 − β2 2 ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ = € 3 −3β1 2 − β₂ 2 EU(fold) = € 1∗β1 2 +1∗ β2 2 +1 1 − β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠

⎟ = 1. For this outcome no calculation is needed since the game ends after player 1 folds. His utility then always is 1. The calculations for player 2’s expected utilities are:

EU(call|H) = € 2 ∗α1 2 + 4 ∗ α2 2 + 3 1 − α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3 −α1 2 + α2 2 EU(call|L) = € 0 ∗α1 2 + 2 ∗ α2 2 + 3 1

(

− α1 2 − α2 2 ⎞ ⎠ ⎟ = € 3 − 3α1 2 − α2 2 EU(fold) = € 1∗α1 2 +1∗ α2 2 + 3 1 − α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3 − α1−α2

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Organizing the expected utilities gives the table below. Player 1 Player 2 EU (bet/call|H) € 3 −β1 2 + β₂ 2 € 3 −α1 2 + α₂ 2 EU (bet/call|L) € 3 −3β1 2 − β₂ 2 € 3 −3α1 2 − α₂ 2 EU (fold) 1 3 - α1 - α2

Table 1 the expected utilities for two risk neutral players

It is striking that the expected utilities are asymmetric for folding. If player 1 folds his utility is always 1 because the game then ends. The expected utility for player 2 depends on his beliefs about player 1 but is at least 1. This implies that there exists a second mover advantage. If player 2 has a low card and wants to fold there is always a chance that player 1 thinks the same. In that case player 2 would get a utility of 3 even though his action was ‘fold’, which would have yielded just 1.

Let’s figure out what the Nash equilibria are in this specific game involving two risk neutral players. Player 1 will play ‘bet’ if his expected utility is

greater than his expected utility from folding. If player 1’s utility from betting is equal to his utility from folding he is indifferent between the two actions. In the case where he has a high card it is optimal to always ‘bet’ since

€

3 −β1

2 + β₂

2 > 1 always holds. However, if player 1 is holding a low card he should ‘bet’ whenever

€

3 −3β1

2 −

β₂

2 > 1. This condition holds always since we considered β1 > β2. Hence, player 1 has a strictly dominating strategy, which is

[BB].

Now let’s do the same for player 2. Player 2’s expected utility from calling when holding a high card should exceed his expected utility from folding. Thus:

€

3 −α1

2 +

α2

2 > 3 - α1 - α2. As becomes clear from this condition it is optimal to ‘call’ with a high card for any value of α1 and α2. This is obvious,

because player 2 either wins or draws when he is holding a high card. If player 2 is holding a low card it is optimal to ‘call’ if

€

3 −3α1

2 −

α2

2 > 3 – α1 – α2. This condition can only hold if α1 is smaller than α2. Since we assumed that

α1 > α2 this condition never holds. Player 2 should therefore always fold when

holding a low card. Just as player 1, player 2 has a strictly dominating strategy, which is [CF].

Each player has four strategies so there are 16 potential Nash equilibria (42₌

16). We have already figured out the optimal actions for any given card, allowing us to determine the Nash Equilibrium(s). As explained above player 1 is best off always playing ‘bet’. This strictly dominating strategy narrows the potential Nash equilibria down to only four.

Player 2’s strictly dominating strategy is to ‘call’ with a high card and to ‘fold’ otherwise. Therefore the only Nash Equilibrium for the belief spaces α1 > α2 and β1 > β2 is [BB, CF]. This Nash equilibrium is illustrated with the

graphs below.

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Player 1 Player 2 1 1 CF β1 BB α1 CC 0 1 0 1 β2 α2

From now assume that α1, α2, β1 and β2 can take on any value on the interval

[0, 1]. Now it is possible for the players to be indifferent between

betting/calling and folding. Does the same Nash equilibrium exist or do new ones arise?

Again, we begin with player 1. When holding a high card it is still optimal to ‘bet’ for any β1 and β2 since

€

3 −β1

2 + β2

2 > 1 always holds. When player 1 is holding a low card it is not always optimal anymore to ‘bet’. For the extreme belief that player 2 calls regardless of his card, that is for β1 = β2 = 1, Player 1 is

now indifferent between betting and folding. EU(bet|L) =

€

3 −3 2 −

1

2 = 1. Thus, for the belief β1 = β2 = 1, player 1 is indifferent between [BB] and [BF].

In contrast to player 1, player 2’s strategy when holding a high card does change. ‘Call’ is no longer strictly dominating ‘fold’. Player 2 is

indifferent if he believes that player 1 never bets, that is α1 = α2 = 0. If player 1

plays according to player 2’s beliefs it does not matter what strategy player 2 chooses since the game then ends before he is allowed to make an action. Besides that, it would be silly to ‘fold’ when holding a high card. When player 2 is holding a low card he is indifferent if he believes that player 1 plays ‘bet’ with a high card with the same probability as ‘bet’ with a low card, that is α1 = α2 = [0, 1]. For all values of α1 > α2 EU(fold|L) > EU(call|L).

So for the belief space α1, α2 =[0, 1] and β1, β2 = [0, 1] there are eight possible

Nash equilibria, which are [BB, CC], [BB, CF], [BB, FC], [BB, FF], [BF, CC], [BF, CF], [BF, FC] and [BF, FF]. Some of them however exist for a very narrow belief space only. In the graphs below the Nash equilibria are listed for which specific beliefs they exist.

Player 1 (BB, BF) Player 2 1 1 β1 α1 (CC, CF 0 1 FC, FF) 1 β2 α1

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Risk averse

Let’s step things up a notch by looking at the optimal strategies for a risk averse and a risk-seeking player. Recall the expected utility functions E(√x) and E(x2_{) for respectively a risk averse and a risk-seeking player.}

Expected utilities for a risk averse player 1: EU(bet|H) = € 2 ∗β1 2 + 4 ∗ β2 2 + 3 1 − β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 2β1 2 +β2+ 3 − 3β1 2 − 3β2 2 = € 3 + β₁ 2 2 ⎛ ⎝ ⎜ − 3 2 ⎞ ⎠ ⎟ + β2 1 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EU(bet|L) = € 0 ∗β1 2 + 2 ∗ β2 2 + 3 1 − β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3 − 3β1 2 +β2 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EU(fold): √1 = 1

For player 2 the expected utilities are: EU(call|H) = € 2 ∗α1 2 + 4 ∗ α2 2 + 3 1 − α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 2α₁ 2 +α2+ 3 − 3α₁ 2 − 3α₂ 2 = € 3 + α1 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EU(call|L) = € 0 ∗α1 2 + 2 ∗ α2 2 + 3 1 − α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3 − 3α1 2 +α2 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EU(fold) = € 1 ∗α1 2 + 1 ∗ α2 2 + 3 1 − α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3 + α1 1 2− 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Player 1 Player 2 EU (bet/call|H) € 3 + β1 2 2 ⎛ ⎝ ⎜ − 3 2 ⎞ ⎠ ⎟ + β2 1 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ € 3 + α1 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EU (bet/call|L) € 3 − 3β1 2 +β2 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ € 3 − 3α1 2 +α2 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EU (fold) 1 € 3 + α1 1 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 2− 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

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Nash equilibria

Just as before player 1 will ‘bet’ if the expected utility from betting exceeds the expected utility from folding. In the case where he has a high card it is

optimal to always play ‘bet’ since

€ 3 + β1 2 2 ⎛ ⎝ ⎜ − 3 2 ⎞ ⎠ ⎟ + β2 1 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ > 1 always holds. If player 1 is holding a low card he should ‘bet’ whenever

€ 3 − 3β1 2 +β2 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠

⎟ > 1. To find the exact values for β1 and β2 for which

this condition holds, we need to find the tipping points of the Nash equilibria. • The first point that has to be considered is (β1, β2) = (0, 0). For these beliefs

the expected utility from betting is larger than the utility from folding. Thus, betting for these beliefs is optimal.

• The second point that needs attention is (β1, β2) = [0, X]. In this case we

consider that β1 = 0 and look for which value(s) of β2 this condition holds.

Up until β2 = 4,61 the condition holds. But since the maximum value of β2

is 1, player 1 plays ‘bet’ for all values of β2 as long as β1 = 0.

• Next, we set β2 = 0 and check β1. If β1 becomes larger than 0,845 the

condition fails. As β2 increases, the value of β1 for which the condition

holds becomes even lower. We know that for β1 > 0,845 the condition

fails no matter what the value of β2 is.

• Now that we have drawn the boundaries when β1 = 0 and β2 = 0 it is time

to look at the intermediate values. To find the intermediate values we can fill in β2 = 1 and set the two equations equal. The two lines intersect at

(0,662; 1), which is the final point that was needed.

• So now we have found the four points for which the condition holds and player 1 will play ‘bet’ when holding a low card. They are: (0, 0), (0, 1), (0,845; 0) and (0,662; 1).

Player 2’s expected utility from calling when holding a high card should exceed his expected utility from folding.

Hence: € 3 + α1 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ > € 3 + α1 1 2− 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . The optimal strategy is identical to the strategy we saw with a risk neutral player. The condition shows that calling with a high card weakly dominates folding. Only for the belief space α1 = α2 = 0 player 2 is indifferent between calling and

folding. Holding a low card it is optimal for player 2 to ‘call’ if

€ 3 − 3α1 2 +α2 2 2 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ > € 3 + α₁ 1 2− 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + α2 1 2− 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . By cancelling out a few terms this condition can be simplified to

€ 2α2 2 > € α1 2 + α2 2 . Similar to player 1, we need to find the tipping points.

• For (0, 0) player 2 is indifferent.

• For α1 = 0 there is no value of α2 for which the condition does not hold.

Hence, up until α2 = 1 the condition is valid.

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• Now, let’s find the intermediate value where α2 = 1. The two lines

intersect at the point (0,414; 1). Any higher value of α1 would make the

condition fail.

• So now we have found the tipping points for which player 2 will be indifferent between playing ‘call’ and ‘fold’ when holding a low card. They are: (0, 0) and (0,414; 1).

So for the belief spaces α1, α2 = [0, 1] and β1,β2 = [0, 1] there exist eight

possible Nash equilibria, which are [BB, CC], [BB, CF], [BB, FC], [BB, FF], [BF, CC], [BF, CF], [BF, FC] and [BF, FF]. The belief space graph below shows for which values these Nash equilibria exist.

Player 1 Player 2 1 1 0,845 β1 (0,662; 1) α1 (BB/BF) (0,414; 1) (CC, CF 0 1 FC, FF) 0 1 β2 α2 Risk seeking

The same calculations have to be made for a risk-seeking player; his utility is now increasing in x.

Expected utilities for a risk-seeking player 1: EU(bet|H) = € 22∗β1 2 + 4 2 ∗β2 2 + 3 2 1 − β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 9 −5β1 2 + 7β2 2 EU(bet|L) = € 02∗β1 2 + 2 2 ∗β2 2 + 3 2 1 − β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 9 −9β1 2 − 5β2 2 EU(fold): 12_{= 1.}

Expected utilities for a risk-seeking player 2: EU(call|H) = € 22 ∗α1 2 + 4 2 ∗α2 2 + 3 2 _{1 −}α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 9 −5α1 2 + 7β2 2 EU(call|L) = € 02 ∗α1 2 + 2 2 ∗α2 2 + 3 2 _{1 −}α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 9 −9α1 2 − 5α2 2

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EU(fold) = € 12 ∗α1 2 +1 2 ∗α2 2 + 3 2 _{1 −}α1 2 − α2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € α₁ 2 + α₂ 2 + 9 − 9α1 2 − 9α2 2 = 9 – 4α1 – 4α2. Player 1 Player 2 EU (bet/call|H) € 9 −5β1 2 + 7β2 2 € 9 −5α1 2 + 7β2 2 EU (bet/call|L) € 9 −9β1 2 − 5β2 2 € 9 −9α1 2 − 5α2 2 EU (fold) 1 9 – 4α1 – 4α2

Table 3 Expected utilities for two risk seeking players Nash equilibria

In the case where player 1 has a high card it is optimal to always ‘bet’ since

€

9 −5β1

2 +

7β2

2 > 1 holds for every value of β1 and β2. If player 1 is holding a low card he should ‘bet’ whenever

€

9 −9β1

2 −

5β2

2 > 1 holds. This is also true for every β1 and β2. Player 1 will always play his strictly dominating strategy [BB].

Player 2’s expected utility from calling when holding a high card should exceed his expected utility from folding. Thus:

€

9 −5α1

2 +

7β2

2 > 9 – 4α1 – 4α2. From this condition it can be derived that calling yields a higher expected utility than folding for any value of α1 and α2 except for the exact belief

€

α1=α2 = 0. Provided that player 2 is holding a low card it is best to ‘call’ if

€

9 −9α1

2 −

5α2

2 > 9 – 4α1 – 4α2. To find the values of the tipping points we perform the same calculations as before.

• For (0, 0) the two equations are equal, which means that player 2 is indifferent for these beliefs.

• If we set α1 = 0, the condition is true for all values of α2.

• If we set α2 = 0, the condition is never met for any positive value of α1.

• This time we have a different intermediate point. Unlike the game with the risk averse players we set α1 = 1 instead of α2 = 1. For any value of

€

α₂ >1

3 the condition holds. So at

€ 1,1 3 ⎛ ⎝ ⎜ ⎞ ⎠

⎟ player 2 is indifferent between calling and folding.

So for the belief spaces α1, α2 =[0, 1] and β1, β2 = [0, 1] there exist four possible

Nash equilibria, which are [BB, CC], [BB, CF], [BB, FC] and [BB, FF].

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Player 1 Player 2 € 1,1 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 1 β1 α1 (CC, CF FC, FF) 0 1 0 1 β2 α2

By comparing the three belief space graphs from the different player types it is interesting to see that the values for which certain Nash equilibria exist shift. For instance, a risk averse player 2 would be less likely to play CC compared to a risk seeking player 2.

The possible Nash equilibria that have been calculated so far are for games where the players are of the same type. In reality two different types of players can face each other. E.g. a risk seeking and a risk averse player or a risk neutral and a risk-seeking player. In total there are nine combinations. It is important to notice that there are two games possible including a risk neutral and a risk averse player, i.e., one game where player 1 is the risk neutral player and the other game where player 1 is the risk averse player. To find all the possible Nash equilibria the calculations as the examples above have to be repeated.

A list of the possible Nash equilibria for all player type combinations is presented in the table below. To find the exact values for which a particular Nash equilibrium exists, one has to replicate the calculations of the examples that are given.

N.B. RN, RA means that a risk neutral player 1 is playing against a risk averse player 2.

RN, RN RN, RA RN, RS RA, RN RA, RA RA, RS RS, RN RS, RA RS, RS

[BB, CC] [BB, CF] [BB, FC] [BB, FF] [BF, CC] [BF, CF] [BF, FC] [BF, FF]

Table 4 Possible Nash equilibria for all combinations of player types using Borel model

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3. Nash equilibria in a two-card Von Neumann poker game

Let’s now switch to an infinite two-card game using the Von Neumann poker model. This game tree is slightly more complicated than the one of the Borel model because there are two outcomes if player 1 plays ‘check’.

Dealer H ( € 1 2) L ( € 1 2) 1 1

Bet Check Check Bet

- - - Dealer - - - H ( € 1 2) L ( € 1 2) H ( € 1 2) L ( € 1 2) H ( € 1 2) L ( € 1 2) H ( € 1 2) L ( € 1 2) 2 2 (2, 2) (3, 1) (1, 3) (2, 2) 2 2 C F C F C F C F (2, 2) (3, 1) (4, 0) (3, 1) (-0, 4) (3, 1) (2, 2) (3, 1)

Figure 4 Game tree of two-card Von Neumann poker model with infinite deck

To be able to find the possible Nash equilibria we formulate the players’ expectations once more.

with 1 – β1)

• If player 1 gets H, player 1 plays ‘bet’ with probability α1 (and ‘check’

with 1 – α1)

• If player 1 gets L, player 1 plays ‘bet’ with probability α2

We consider that β1, β2, α1 and α2 can have any value on the interval [0, 1].

To find the optimal strategies for each player we practically have to cover the same steps as with the Borel model. There are however a few minor

differences. Both players now have to compare their expected utility from betting/calling with a high card with checking/folding with a high card. The same applies for holding a low card. The expected utility for player 2 now consists of four parts instead of three.

The expected utility of player 2 consists of:

- The utility if player 1 plays bet with H, times the subjective probability that player 1 plays bet.

- The utility if player 1 plays bet with L, times the subjective probability that player 1 plays bet.

- The utility if player 1 plays check with H, times the subjective probability that player plays check.

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- The utility if player 1 plays check with L, times the subjective probability that player plays check.

Player 1 can now ‘check’ instead of ‘fold’ if he does not want to play ‘bet’. If this happens the cards of both players will be compared. Player 2 then is not allowed to make an action. If player 1 chooses ‘check’ his expected utility consists just of these two parts:

- The utility if player 2 has a high card, times the probability that player 2 has a high card. In our game we assumed that this probability is equal to a half.

- The utility if player 2 has a low card, times the probability that player 2 has a low card.

Below an example is given for the calculations that have to be made to be able to find the optimal strategies with their corresponding Nash equilibria.

Nash equilibria

Player 1 plays ‘bet’ if his expected utility from betting exceeds his expected utility from checking. In formula:

€ 3 −β1 2 + β₂ 2 > € 5

2. Only for the belief space where (β1, β2) = (1, 0) player 1 is indifferent. For any other belief space player 1

plays ‘bet’.

The condition that must hold for player 1 to play ‘bet’ with a low card is:

€ 3 −3β1 2 − β₂ 2 > € 3

2. This condition can be simplified to

€ 3β₁ 2 + β₂ 2 < 3 2.

• Same as with a high card, (0, 0) is supported. For β1 = 0 the condition is

true for β2 < 3. The maximum value of β2 is 1; so all values up until 1

give the optimal strategy ‘bet’.

• For β2 = 0 the condition holds for all values of β1 until 1. The point (1, 0)

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• Finally, the intermediate point has to be computed. Fill in β2 = 1 and set

the two equations equal. The equation is valid as long as β1 does not

exceed

€

2

3. The indifference point is

€ 2 3,1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ .

• The two boundary points for player 1 are (1, 0) and

€ 2 3,1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . For player 2 the condition

€ 13 2 + 7α2 2 > 13 2 − 3α1

2 − 4α2 must hold in order for him to play ‘call’ with a high card. Only for the belief space

€

α1=α2 = 0 player

2 is indifferent. For all other values of α1 and α2 player 2 chooses ‘call’. If

player 2 is holding a low card he calls if

€ 5 2 − α1 2 > 5 2 − 3α2 2 . • For € α1,α2

(

)

= 0,0

( )

player 2 is indifferent.

• If we set α1 = 0 we find that for every positive value of α2 the condition

holds.

• For α2 = 0 the condition always fails if α1 has a nonzero value.

• The intermediate point can be found by setting α1 = 1. We find that

once α2 becomes greater than

€

1

3 the condition holds, that is player 2 will play ‘call’.

• The indifference points for player 2 are: (0, 0) and

€ 1,1 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ .

So for the belief spaces α1, α2 =[0, 1] and β1, β2 = [0, 1] there exist 16 Nash

equilibria. In the belief space graphs below the Nash equilibria are shown. Notice that 12 of the possible equilibria exist for a very small belief space.

Player 1 Player 2 (BB, BC, CB, CC) € 1,1 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 1 β1 € 2 3,1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ α1 (CC, CF 0 1 FC, FF) 0 1 β2 α2

Just as with the Borel model there are nine combinations of player types that can play against each other. The calculations for the Nash equilibria are comparable to the given example so it is unnecessary to show them all. It is striking that for six of the nine combinations 16 Nash equilibria are possible. However, 12 of these equilibria exist only on the narrow belief space where (α1, α2) = (0, 0) and (β1, β2) = (1, 0). Table 6 below provides all possible Nash

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RN, RN RN, RA RN, RS RA, RN RA, RA RA, RS RS, RN RS, RA RS, RS [BB, CC] [BB, CF] [BB, FC] [BB, FF] [BC, CC] [BC, CF] [BC, FC] [BC, FF] [CB, CC] [CB, CF] [CB, FC] [CB, FF] [CC, CC] [CC, CF] [CC, FC] [CC, FF]

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4. A bridge from Borel to Von Neumann

In reality, a poker player is not restricted to bet the same amount every time. In fact, there are countless amounts that can be used to bet with. The next model that will be introduced, designed by Bellman and Blackwell (1949), can be seen as a bridge between the Borel and Von Neumann poker models. In this model Bellman and Blackwell allow player 1 to choose between two bet sizes. That is, after receiving his card, player 1 may fold, or bet B1, or bet B2,

where 0 ≤ B1 ≤ B2. In the case where B1 = B2 this is exactly Borel’s model. In the

case where B1 = 0 this is comparable to Von Neumann’s model, since player 1

would never fold if he can bet 0, which is equivalent to checking. For player 2 all actions remain the same.

Strategy profile for both players:

Player 1: {bet1, bet2, fold} where we consider bet1 < bet2.

Player 2: {call, fold}

In the game tree below the ante is 1, B1 = 1 and B2 = 2.

Dealer High (½) Low (½) (2, 4) (2, 4) Fold 1 1 Fold B1 B2 B1 B2 2 - - - 2 Dealer 2 - - - 2 H L H L H L H L C F C F C F C F C F C F C F C F (3, 3) (4, 2) (3, 3) (4, 2) (1, 5) (4, 2) (0,6) (4, 2) (5, 1) (4, 2) (6, 0) (4, 2) (3, 3) (4, 2) (3, 3) (4, 2) Figure 5 Game tree of a Bellman and Blackwell poker game with infinite deck

To be able to find the possible Nash equilibria we formulate the players’ expectations.

with 1 – β1)

• If player 1 gets H, player 1 plays ‘B1’ with probability α1H

• If player 1 gets H, player 1 plays ‘B2’ with probability α2H (and ‘fold’

with 1 – α1H – α2H)

• If player 1 gets L, player 1 plays ‘B1’ with probability α1L

• If player 1 gets L, player 1 plays ‘B2’ with probability α1L (and ‘fold’

with 1 – α1L – α2L)

We assume that α1H + α2H, α1L + α2L and β1 and β2 can take on any value on the

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Example: risk-seeking player 1 vs. risk neutral player 2

Now, this is where things get really complicated. Player 1 is allowed to

choose from three actions. But since player 1 can be dealt a high card or a low card we have to multiply his possible actions by two. However, player 2 has to consider ‘just’ five actions, because when player 1 folds it is irrelevant with what card he folds.

The expected utility of player 2 therefore consists of:

- The utility if player 1 plays B1 with H, times the subjective probability

that player 1 plays B1

- The utility if player 1 plays B2 with H, times the subjective probability

- The utility if player 1 plays B1 with L, times the subjective probability

- The utility if player 1 plays B2 with L, times the subjective probability

- The utility if player 1 plays fold, times the subjective probability that player 1 plays fold

Expected utility for a risk neutral player 2: EU(call|H) = € 3∗α1H 2 + 3∗ α2 H 2 + 5 ∗ α1L 2 + 6 ∗ α2L 2 + 4 1 − α1H 2 − α2 H 2 − α1L 2 − α2L 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 3α1H 2 + 3α2 H 2 + 5α1L 2 + 3α2L+ 4 − 2α1H − 2α2 H − 2α1L − 2α2L = € 4 −α1H 2 − α2 H 2 + α1L 2 +α2L EU(call|L) = € 1∗α1H 2 + 0 ∗ α2 H 2 + 3∗ α1L 2 + 3∗ α2L 2 + 4 1 − α1H 2 − α2 H 2 − α1L 2 − α2L 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € α1H 2 + 3α1L 2 + 3α2L 2 + 4 − 2α1H − 2α2 H − 2α1L − 2α2 H = € 4 − 3α1H 2 − 2α2 H − α1L 2 − α2L 2 EU(fold) = € 2 ∗α1H 2 + 2 ∗ α2 H 2 + 2 ∗ α1L 2 + 2 ∗ α2L 2 + 4 1 − α1H 2 − α2 H 2 − α1L 2 − α2L 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 4 − α1H −α2 H −α1L −α2L

The expected utility for player 1 still consists of these three parts:

- The utility if player 2 plays call with H, times the subjective probability that player 1 plays call.

- The utility if player 2 plays call with L, times the subjective probability that player 1 plays call.

- The utility if player 2 plays fold, times the subjective probability that player 1 plays fold.

There are some changes for player 1 nevertheless. In the Borel and Von

Neumann models player 1 has to compare his expected utility from betting to his expected utility from folding or checking. In this model player 1 has to compare three expected utilities. He has to compare his expected utility from betting B1 with his expected utility from betting B2 and his expected utility

from folding.

The utility that each player derives from a draw changes as well. If player 1 bets and player 2 calls and they have the same card the utility is now 3 instead of 2. The maximum utility a player can get now is 6.

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Expected utility for a risk-seeking player 1: EU(B1|H) = € 32 ∗β1 2 + 5 2 ∗β2 2 + 4 2 _{1 −}β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 16 −7β1 2 + 9β2 2 EU(B2|H) = € 32 ∗β1 2 + 6 2 ∗β2 2 + 4 2 _{1 −}β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 16 −7β1 2 +10β2 EU(B1|L) = € 12 ∗β1 2 + 3 2 ∗β2 2 + 4 2 _{1 −}β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 16 −15β1 2 − 7β2 2 EU(B2|L) = € 02 ∗β1 2 + 3 2 ∗β2 2 + 4 2 _{1 −}β1 2 − β2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = € 16 − 8β1− 7β2 2 EU(fold) = 22_{= 4} Player 1 Player 2 EU(B1|H) € 16 −7β1 2 + 9β2 2 EU(B2|H) € 16 −7β1 2 +10β2 EU(B1|L) € 16 −15β1 2 − 7β2 2 EU(B2|L) € 16 − 8β1− 7β2 2 EU(call|H) € 4 −α1H 2 − α_{2 H} 2 + α_1L 2 +α2L EU(call|L) € 4 − 3α1H 2 − 2α2 H − α1L 2 − α2L 2 EU(fold) 4 € 4 −α1H −α2 H −α1L −α2L Table 7 Expected utilities for risk-seeking player 1 and risk neutral player 2 Nash equilibria

Once more, the expected utility for player 1 is he bets B1 must exceed his

expected utility from folding. The condition

€

16 −7β1

2 +

9β2

2 > 4 holds for all values of β1 and β2. The same holds for the condition

€

16 −7β1

2 +10β2 > 4. The question that remains is whether player 1 should bet B1 or B2. To see which

bet size yields a higher expected utility the expected utilities of these two actions have to be compared. The condition

€ 16 −7β1 2 +10β2 > € 16 −7β1 2 + 9β2 2 holds for all values of β2 > 0. For β2 = 0 player 1 is indifferent between B1 and

B2. If player 1 has a low card he is also better off betting since his expected

utilities from B1 and B2 both exceed his expected utility from folding. This

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Player 2’s optimal strategy when holding a high card is calling since € 4 −α1H 2 − α2 H 2 + α1L 2 +α2L > €

4 − α_1H −α_{2 H} −α_1L −α_2L is true for any value of α1H,

α2H, α1L and α2L. Only for the extreme belief that player 1 folds with any card

player 2 is indifferent between calling and folding. In the case where player 2 is holding a low card it depends on his beliefs about player 1 whether he should call or fold. From the condition

€ 4 − 3α1H 2 − 2α2 H − α1L 2 − α2L 2 > € 4 −α1H −α2 H −α1L −α2L it is not immediately

clear for what values this condition holds. Player 2 for instance has the following beliefs: € α1H = 4 10; € α2 H = 6 10; € α1L = 1 10; € α2L = 1 10. Player 2’s expected utility for calling for these beliefs is:

€ 4 −3 2∗ 4 10− 2 ∗ 6 10− 1 2∗ 1 10− 1 2∗ 1 10 = € 21 10. Folding for these beliefs is better since

€ 4 − 4 10− 6 10− 1 10 − 1 10 = € 28 10 > 21 10. If player 2 however has the beliefs

€ α1H = 6 10; € α2 H = 1 10; € α1L = 1 2; € α2L = 1 2 his expected utility for calling is:

€ 4 −3 2∗ 6 10 − 2 ∗ € 1 10 – € 1 2∗ 1 2− 1 2∗ 1 2 = € 24 10. This is larger than his expected utility for folding, which is

€

23

10. Now player 2 will play call.

For the belief spaces α1H + α2H, α1L + α2L =[0, 1] and β1, β2 = [0, 1] there

exist 16 Nash equilibria, which are [B1B1, CC], [B1B1, CF], [B1B1, FC], [B1B1, FF],

[B1B2, CC], [B1B2, CF], [B1B2, FC], [B1B2, FF], [B2B1, CC], [B2B1, CF], [B2B1, FC],

[B2B1, FF], [B2B2, CC], [B2B2, CF], [B2B2, FC] and [B2B2, FF].

With the current assumptions, it is quite complicated to figure out for which value(s) certain equilibria exist. So to be able to find the exact values of α1H,

α2H, α1L and α2L we make a simplifying assumption. We assume that player 2

knows that player 1 never bluffs big, i.e., player 1 never bets B2 if he is holding

a low card. What this simplifying assumption reveals is that whenever player 1 bets B2, player 2 knows that player 1 has a high card. The α2L part of the

expected utility function for player 2 disappears. The expected utility functions now look like this:

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Nash equilibria

With this extra assumption there are just two strategies that player 1 will use. If he is holding a low card he now has the strictly dominating strategy, which is to bet B1. Player 1 will either play [B1B1] or [B2B1]. For all nonzero values of

€

β2 he will play the latter.

Player 2’s optimal strategy has to be computed in the same way as the strategies from previous chapters. If player 2 is holding a high card his optimal strategy is to call for all positive values of α1H, α2H and α1L since then

€ 4 −α1H 2 − α_{2 H} 2 + α_1L 2 > €

4 −α1H −α2 H −α1L. If player 2 is dealt a low card he calls

when € 4 − 3α1H 2 − 2α2 H − α_1L 2 > €

4 −α1H −α2 H −α1L. We now need to find the

tipping points in order to say something meaningful about the existence of particular Nash equilibria.

• We know that for (α1H, α2H, α1L) = (0, 0, 0) player 2 is indifferent.

• Now, we let (α1H, α2H, α1L) = (0, 0, X). For any nonzero value of α1L

player 2 will call.

• Next, we let (α1H, α2H, α1L) = (0, X, 0). Opposite of α1L player 2 will now

fold for any nonzero value of α2H.

• And now let (α1H, α2H, α1L) = (X, 0, 0). Same as for α2H, every nonzero

value of α1H will make player 2 fold.

• Now we can look for values of α1H and α2H where α1L is assumed to be

1. If we substitute α1L = 1 in the condition we get:

€ 7 2 − 3α1H 2 − 2α2 H > €

3 −α1H −α2 H. If we let α2H = 0, then α1H < 1 will give

the optimal action ‘call’. For α1H = 1 player 2 is indifferent.

• When we let α1H = 0 for every value of α2H <

€

1

2 player 2 will ‘call’. • Finally, for every combination of α1H andα2H for which

€

α_1H

2 +α2 H < 1 2 holds player 2 plays ‘call’.

• To summarize, we found these points for which player 2 is indifferent between calling and folding: (0, 0, 0), (0,

€ 1 2, 1), (1, 0, 1) and (α1H, α2H, 1) for which € α1H 2 +α2 H < 1 2.

Hence, a Bellman and Blackwell game involving a risk seeking player 1 and a risk neutral player 2 has 8 possible Nash equilibria. These are, together with all other player types combinations, listed in table 9 on the next page.

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RN, RN RN, RA RN, RS RA, RN RA, RA RA, RS RS, RN RS, RA RS, RS [B1B1, CC] [B1B1, CF] [B1B1, FC] [B1B1, FF] [B2B1, CC] [B2B1, CF] [B2B1, FC] [B2B1, FF] [B1F, CC] [B1F, CF] [B1F, FC] [B1F, FF] [B2F, CC] [B2F, CF] [B2F, FC] [B2F, FF]

Table 9 Possible Nash equilibria for all combinations of player types excluding the (B2|L) action for

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5. Conclusion

Currently several poker models exist that try to approximate the optimal strategies in a simplified game of poker. This thesis attempts to answer the question “What are the Nash equilibria in a two-card poker game using the Borel and Von Neumann poker models as a guideline?”

In chapters two and three the Borel and Von Neumann models are used to find the possible Nash equilibria in a two-card poker game. In the fourth chapter of the thesis these two models are combined.

Using the Borel model we found the optimal strategies for every combination of the three player types. Eight Nash equilibria exist in a game that involves either a risk neutral or risk averse player 1. In the remaining three games with a risk-seeking player 1 there are four Nash equilibria. What we also found is that the more risk loving the players are, the more the

players tend to play ‘bet’ or ‘call’ with a low card.

The results for the Von Neumann model are obviously different. The number of possible Nash equilibria doubled compared to the Borel model. In a game involving either a risk neutral or a risk seeking player 1 twelve of these Nash equilibria exist only on the narrow belief space

€

α1,α2

(

)

= 0,0

( )

and

€

β1,β2

(

)

= 1,0

( )

. Unlike the Borel model, player 1 does not have a strictly dominating strategy when holding a high card. The optimal strategies for player 2 remain unchanged.

In the final chapter we learned that the number of Nash equilibria increases even more in a game where player 1 is allowed to choose between two bet sizes. After restricting the action (B2|L) for player 1 the possible Nash

equilibria narrowed down to 16. When holding a low card, a risk averse player 1 has the tendency to fold for certain high values of β1 and β2 whereas a

risk neutral and a risk-seeking player 1 always bet. In all three models player 2’s strategy remained constant. This can be explained by the fact that player 2’s action profile consisted of the same two choices throughout the models.

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6. Review

All the results in this thesis rely on a few assumptions. Firstly, this thesis assumes that both players have to contribute an ante of 1 unit in advance of the game. In real life poker only the ‘small blind’ and ‘big blind’ are obliged to put an ante in the pot. In the Borel model this has a negative impact on player 1. If player 1 would not have been obliged to contribute an ante the optimal strategies for player 1 might have been different. The same applies for player 2 in the Von Neumann model.

An infinite deck is another assumption made in this thesis. This is by no means a correct representation of reality. In a full-scale game of poker players are aware of the amount of cards in the deck. Due to this, a skilled poker player is able to calculate his winning chances by comparing the cards on the table with his hand. This important factor vanishes in the two-card game with an infinite deck. A possible extension of this thesis could be to determine the Nash equilibria in a game that involves so-called ‘odds’ and ‘pot odds’, i.e., a game in which players are able to calculate the chance of getting a particular card that is needed to win the pot.

This thesis only tackles a two-card game that is played between two players. A full-scale game of poker is of course played with 13 different cards and more players. The results therefore lack external validity. An interesting extension of this thesis might be to look for the Nash equilibria in a 3, 4 or 5 card poker game. It will be more complicated to find the Nash equilibria but the results will be more valuable. Another idea for future research is to find the optimal strategies in a poker game involving three or four players. E.g. extend the work of John Nash, who wrote a small piece about a three-man poker game in his book ‘The Annals of Mathematics’ in 1951.

Furthermore, in the poker game that is being played in this thesis the players interact only once. Normally a poker game is played over an extended time frame until just one player remains. This game therefore differs from reality to the extent that players cannot take their opponents’ strategies into account from the previous games. In real-world poker a player might want to change his strategy after losing a couple of times in a row or to be

unpredictable. Future research could extend this work by investigating the Nash equilibria in a two-shot game.

Lastly, in chapter 4 player 1 was allowed to choose between two bet sizes. For this reason, the Bellman and Blackwell model is slightly more realistic than the Borel and Von Neumann models. A full-scale game of poker allows all players to bet a continuous amount. This has an impact on the opponents’ actions. Betting a large amount while having a bad hand may be a good strategy since your opponents may think you truly have a good hand.

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References

Billings, D., Burch, N., Davidson, A., Holte, R., Schaeffer, J., Schauenberg, T., Szafron, D. Approximating game-theoretic optimal strategies for full-scale poker. Proceedings of IJCAI-03, (Eighteenth International Joint Conference on Artificial Intelligence), 2003.

Ferguson, Thomas, and Ferguson, Chris. (2003) “On the Borel and von

Neumann Poker Models,” Game Theory and Applications 9 (2003), 17-32. Gilpin, A., Sandholm, T. Better automated abstraction techniques for

imperfect information games, with application to Texas Hold’em poker. The Sixth Intl. Joint Conf. on Autonomous Agents and Multi-Agent

Systems (AAMAS 07).

Global interactive poker gross win from 2004 to 2015 (in billion euros) acquired

from http://www.statista.com/statistics/208468/global-interactive-poker-gross-win-since-2004/ on 23rd_{of January 2016.}

How to play | Terms acquired from http://www.wsop.com/poker-terms/ on

3rd_{of November 2015.}

Osborne, M (2009), “An Introduction to Game Theory”, Oxford University Press, New York

Nash, J (1951). Non-cooperative games. The Annals of Mathematics, second series, Volume 54, issue 2 (Sep., 1951), 286-295.

Von Neumann, Morgenstern. Theory of Games and Economic Behaviour, Princeton University Press, 1944.

The Nash equilibria in a two-card poker game