summer paper - terrence adam rooney
TRANSCRIPT
A Game-Theoretic Analysis of a Poker Hand
Terrence Adam Rooney1
September 30, 2016
Abstract
This paper presents a model motivated by the first betting round in a single, two -
player, No-Limit Hold’em poker hand. In this model, the two, risk-neutral, players
engage in a static, simultaneous-move game of incomplete information that
incorporates luck along with technical and perceptive skill. Through a comparative
static analysis, I develop insights regarding the parametric effects of the model.
Through this analysis, the insights I reveal are consistent with conventional poker
wisdom and the results obtained by other works of literature.
1 I would like to thank Professor Peter Streufert for the insight he has provided to me throughout the development of this paper. I would also l ike to thank Professor Gregory Pavlov, along with my classmates, Minku Kang and
Shannon Potter, for their helpful comments.
1 Introduction & Related Literature
As the internet grew to prominence in the early 21st century, many industries
were revolutionized and, consequently, experienced a drastic change in prevalence
and popularity. The poker industry was no exception. After Chris Moneymaker, a
professional accountant and amateur online poker player, won the 2003 World Series
of Poker “WSOP” Main Event, the number of entrants in this tournament grew from
839 in 2003 to 5,619 in 2005 (Croson et al., 2008). People rationalize that, if Chris
Moneymaker could qualify online for the WSOP Main Event and win, then anyone
can do so (McCormack & Griffiths, 2012). Over the past decade, the large number of
entries to this World Championship tournament has persisted, with the most recent
event drawing a field of 6,737 players (“WSOP.com”, 2016). Additionally, it is
estimated that approximately eighty million people (over 1 percent of the global
population) play poker on a regular basis (Holden, 2008).
Despite the significant growth in popularity that poker has experienced, very
little empirical research regarding online poker has been conducted (Griffiths et al.,
2010). However, there have been various attempts to theoretically devise an optimal
poker strategy. Burns (2004) examines the roles of style and skill in a simplified poker
game, and shows that the best style for one player depends on the style of her
opponent. Burns (2004) also demonstrates, in his model, that Novice styles can be
remarkably effective against Expert skill. Although initially unintuitive, the logic
behind this conjecture is based on the fact that an expert poker player is able to
capitalize on predicting their opponent’s behaviour, and hence gains a competitive
advantage. But, when an expert player is facing a novice opponent that does not know
how to optimally, or even skillfully, play their hand, the expert’s advantage is
mitigated since the novice’s approach will be more random and unpredictable than
that of a moderately skilled player.
Dreef et al. (2004) also attempt to devise an optimal poker strategy by
providing an overview of the relevant aspects of a method, presented by Borm and
Van der Genugten (2000, 2001), that determines whether a game can be classified as
a game of skill or not. Dreef et al. (2004) utilize a simple, two-person poker game to
illustrate the concepts presented in their paper. However, they still have some
difficulty with incorporating skill concepts in game theory (McCormack, A., &
Griffiths, 2012). The model that I present in this paper also features a simple, two-
person poker game, similar to that which is contained in Dreef et al. (2004). Both of
our models assume that players of a higher ability have better information than their,
lesser-skilled, counterpart. My paper, however, will split a player’s skill into two
ability components, informational (technical) and perceptive, whereas Dreef et al.
(2004) assumes that players only differ in their informational (technical) skill.
Another critical difference between our models is that Dreef et al. (2004) features a
sequential-move game, whereas my model presents a simultaneous-move game.
Although, in reality, poker is a sequential-move game, I have chosen to implement a
simultaneous-move structure since it presents the possibility of applying my results
to many real-world situations, such as entrepreneurs (or firms) entering an industry
and job-seekers searching for employment. I will address these applications, among
others, in a future paper, and have decided, here, to limit my focus to presenting my
model and conducting a comparative static analysis on each of the model’s features.
Poker has been researched in many fields of study, such as computer science,
economics, psychology, and statistics, with the majority of academic research taking
place in fields outside of economics. For instance, works from the computer science
field include Billings et al. (2002, 2003), Korb et al. (1999), Shi & Littman (2000), and
Southey et al. (2012); works from the psychology field include Griffiths et al. (2010),
McCormack & Griffiths (2012), and Rapoport et al. (1997); and works from the
statistics field include Borm and Van der Genugten (2000, 2001), and Crosen et al.,
(2008).
Research papers that have linked economics with poker primarily fall within
the field of Behavioral Economics or focus on the empirical relationship between luck
and skill. Siler (2010) analyzes poker hands played online in small, medium, and high
stakes, then determines which strategies are conducive to winning at each level.
Consequently, Siler (2010) finds that a skillful player’s competitive edge diminishes
as she moves up levels, while at the same time, tight-aggressive strategies, which
tend to be the most lucrative, become more prevalent.
Levitt & Miles (2014) apply data from the 2010 WSOP to shed insight on
whether poker is a game of luck or skill. Their results provide strong evidence that
supports the idea that poker is a game of skill. This is because, in their study, players
identified as highly skilled prior to the start of the 2010 WSOP finished with a return
on investment of over 30%, whereas all other players combined achieved a -15%
return on investment. Hannum & Cabot (2009) arrive at a similar result as Levitt &
Miles (2014), in the sense that they also find that poker is predominately a game of
skill. To reach this conclusion, Hannum & Cabot (2009) run a mathematically-based
simulation study that demonstrates the payout benefits for a player that employs a
skillful strategy versus the drawbacks for a player utilizing a randomized (unskilled)
strategy. Contrarily, Meyer et al. (2013) finds that poker, under basic conditions,
should be regarded as a game of chance, since their findings indicate that the
outcomes of poker games are predominantly determined by chance. Meyer et al.
(2013) arrive at this conjecture by completing a quasi-experimental study that
examines the extent to which poker skill was more important than the distribution
of cards dealt. Upon taking this all into consideration, it is clear that the luck versus
skill debate with poker has yet to be settled.
Although the model that I feature in this paper is meant to replicate a
simplified poker game, it has various similarities to other theoretical models and
concepts detailed in economics literature. Haltiwanger & Waldman (1985)
investigates a model that emphasizes heterogeneity between agents that vary in
terms of informational processing ability. They find that in an environment
characterized by congestion effects (i.e. the higher the number of agents choosing a
strategy, the worse off each corresponding agent is), informationally-skilled agents
have a disproportionally large effect on equilibrium. Their findings also indicate that
in an environment characterized by synergistic effects, naïve agents tend to be
disproportionally important. Aumann & Heifetz (2002) acknowledges the importance
of incorporating a player’s beliefs about the other players’ beliefs into game theoretic
models, and it also details various ways of doing this. Finally, Jehiel & Koessler
(2008) studies the effects of analogy-based expectations in static two-player games of
incomplete information. Their paper assumes that players only understand the
average behaviour of their opponent over bundles of states, then characterizes
players by how finely they understand the strategy of their opponent together with
their own information and payoff structure.
Finally, in addition to the above research, a related stream of economics
literature is that of overconfidence, and other personality traits, in contests. Ando
(2004) investigates an economic contest featuring two players who are each
overconfident in their own relative abilities. Specifically, Ando (2004) studies two
different sources of overconfidence, a player’s overestimation of their own ability and
a player’s underestimation of their opponent’s ability. He finds, in his paper, that a
player’s overestimation of their own ability leads to that player exhibiting aggressive
behaviour, whereas a player’s underestimation of their opponents’ ability sometimes
leads to less aggressive behaviour by one or both players. Therefore, according to this
result, overconfidence may not always lead to aggressive play. Ludwig et al. (2011)
utilize a model that incorporates a two-player Tullock contest to show that modest
overconfidence in a contest can improve a player’s performance relative to an
unbiased opponent, and possibly lead to an absolute advantage for the overconfident
player. Finally, through an experimental study, Alaoui & Fons-Rosen (2016)
examines how grit, which is linked to perseverance, influences behaviour. They show
that along with this commonly known upside of grit, there is also a potential downside
in the form stubbornness.
The remainder of my paper is as follows. Section 2 presents the motivation
behind my model. Section 3 details the model. Section 4 explains the parameters that
I have chosen for the model’s benchmark case. Section 5 presents the analysis, and
Section 6 concludes.
2 Motivation
The model I will present is motivated by the first betting round of a single, two-
player, hand of No-Limit Hold’em poker. In a standard hand, players are each dealt
two cards, and are endowed with a set number of chips (i.e. capital). In order to
incentivize betting, players are obligated to pay blinds into the pot. Players then
proceed through playing the hand2.
A common occurrence in tournament poker is when the blinds are large in
relation to each player’s number of chips. When this happens, a popular strategy , in
the first betting round, for players is to either fold their hand and relinquish the blind
they have paid into the pot, or raise (henceforth known as call) all of their chips and
put maximum pressure on their opponent to fold, which would result in the player
winning the blind of their opponent. With this in mind, I have decided to conduct
analysis based on the assumption that each player can choose one of only these two
actions.
2 Please refer to “Texas Hold’em Rules” (2016) for an overview of how a poker hand is played.
3 Model
I consider a model that features two, risk-neutral, players engaging in a static,
simultaneous-move game of incomplete information. Each player (𝑖 ∈ {1, 2}) is
endowed with an initial level of capital, 𝑘𝑖0, and there exists a blind cost (𝑏) such that
𝑏 ∈ (0,𝑀𝑖𝑛{𝑘10,𝑘2
0}]. Each player (𝑖 ∈ {1, 2}) is endowed with an informational ability
coordinate, (𝑥𝑖 ,𝑦𝑖), that satisfies the following constraints:
𝑥𝑖 ∈ [0,1] and 𝑦𝑖 ∈ [0,1] (1) − "𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦"
1
2(1 − 𝑥𝑖) + 𝑦𝑖 ≤ 1 (2) − "𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑖𝑡𝑦"
4𝑦𝑖 − 𝑥𝑖 ≥ 1 (3) − "𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒"
Additionally, each player (𝑖 ∈ {1, 2}) is assigned a perceptive ability3 𝑝𝑖 ∈ {1,4}.
3.1 Timeline
The timeline for this game is as follows. For each player (𝑖 ∈ {1, 2}), Nature
draws, with equal probability, a hand-type, 𝑐𝑖 ∈ {1, 2, 3}, as well as a lottery-type, Λ𝑖 ∈
{𝐿1(𝑥𝑖 ,𝑦𝑖),𝐿2(𝑥𝑖 ,𝑦𝑖),𝐿 3(𝑥𝑖 ,𝑦𝑖)}. Each player (𝑖 ∈ {1, 2}) observes their own abilities,
lottery-type, capital values, blind cost, their opponent’s perceptive ability, and if 𝑝𝑖 =
1, their opponent’s informational ability. Upon making these observations, each
player (𝑖 ∈ {1,2}) chooses an action, 𝑎𝑖(Λ𝑖), such that 𝑎𝑖(Λ𝑖) ∈ {Fold, Call}4. For 𝑖 ∈
{1, 2}, if player 𝑖 chooses “Fold”, the blind cost (𝑏) is deducted from player 𝑖′𝑠 capital,
3 Both ability types for player 𝑖 (𝑖 ∈ {1, 2}) are independent of each other. Additionally, each ability type for player
𝑖 is independent of the ability types endowed to player −𝑖. 4 In the Analysis section, “Fold” will be denoted as “F”, and “Call” will be denoted as “C”.
and added to player −𝑖′𝑠 capital (𝑖.𝑒. ∶ 𝑘𝑖 = 𝑘𝑖0 − 𝑏 & 𝑘−𝑖 = 𝑘−𝑖
𝑜 + 𝑏). If both players
choose “Call”, then the following showdown occurs. For 𝑖 ∈ {1,2}, Nature draws three
additional values, 𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 ∈ {1,2, 3}5. Subsequently, for 𝑖 ∈ {1, 2}, the capital values
update as follows:
𝑘𝑖 = {
𝑘𝑖0 + 𝑀𝑖𝑛{𝑘𝑖
0, 𝑘−𝑖0 } 𝑖𝑓 𝑐𝑖 + 𝜆𝑖 > 𝑐−𝑖 + 𝜆−𝑖
𝑘𝑖0 − 𝑀𝑖𝑛{𝑘𝑖
0, 𝑘−𝑖0 } 𝑖𝑓 𝑐𝑖 + 𝜆𝑖 < 𝑐−𝑖 + 𝜆−𝑖
𝑘𝑖0 𝑖𝑓 𝑐𝑖 + 𝜆𝑖 = 𝑐−𝑖 + 𝜆−𝑖
(4)
Where:
𝜆𝑖 = 𝛼𝑖 + 𝛽𝑖 + 𝛾𝑖 (5)
Finally, payoffs are realized for 𝑖 ∈ {1, 2}, such that:
𝑈𝑖 (𝑘𝑖; 𝑎𝑖 , Λ𝑖 , 𝑐𝑖 , 𝑝𝑖 , 𝛼𝑖 ,𝛽𝑖 , 𝛾𝑖 ) = 𝑘𝑖 (6)
3.2 Features of the Model
The main innovation of my model is that I incorporate two different types of
skill (technical and perceptive), as well as luck, into a static, simultaneous move, two-
player game of incomplete information. In this subsection, I will present an overview
of both skill types, and the luck component, of my model.
3.2.1 Technical Skill
The informational ability (technical skill) coordinate that I incorporate is
inspired by the incomplete information about alternatives, bounded rationality,
5 Each additional value has a
1
3 chance of being a 1, a
1
3 chance of being a 2, and a
1
3 chance of being a 3.
concept explained in Simon (1972). With this ability coordinate, I impose, on each
player, incomplete information regarding the hand-type that each player is initially
dealt. Upon receiving the, initially, unobservable hand-type, the player’s goal is to
choose the alternative that maximizes their expected payoff (Simon, 1972). The
ability coordinate acts as a proxy for a player’s level of technical skill in poker. The
lottery-types that a player may draw are computed using that player’s informational
ability coordinate, and they represent that player’s ability to discern the true value
of their hand-type. Specifically, the 𝑥𝑖 coordinate represents player 𝑖′𝑠 capability of
discerning the middling hand-type, 𝑐𝑖 = 2, while the 𝑦𝑖 coordinate represents player
𝑖′𝑠 skill of discerning the extreme hand-types, 𝑐𝑖 = 1 and 𝑐𝑖 = 3.
As mentioned above, players observe their lottery-type, however, they do not
observe their hand-type. Each lottery-type, that player 𝑖 may draw, is a convex
combination of hand-types, where the coefficients represent the probability that
player 𝑖 has drawn the corresponding hand-type. For instance, if player 𝑖 draws a
lottery-type that is the convex combination 0.75[1] + 0.125[2] + 0.125[3], player 𝑖
would know that there is a 75% probability they drew 𝑐𝑖 = 1, a 12.5% probability they
drew 𝑐𝑖 = 2, and a 12.5% probability they drew 𝑐𝑖 = 3.
To coincide with the fact that every convex combination must have non-
negative coefficients that sum to 1, I impose, for convenience, two additional
assumptions that will allow me to write a system of equations, with two degrees of
freedom, for all lottery-type coefficients. The first assumption is that the coefficients
for each hand-type, in all of the lottery-type convex combinations, will sum to 1. This
assumption ensures that each player will have an equal probability of drawing every
specific hand-type, regardless of the outcome of their draw of the informational ability
coordinate. The second assumption is symmetry between the lottery-types.
Specifically, the symmetry I am imposing is that:
I. The probability coefficient for 𝑐𝑖 = 1 in 𝐿1(𝑥𝑖 ,𝑦𝑖) is equal to the
probability coefficient for 𝑐𝑖 = 3 in 𝐿 3(𝑥𝑖 ,𝑦𝑖)
II. The probability coefficient for 𝑐𝑖 = 1 in 𝐿2(𝑥𝑖 ,𝑦𝑖) is equal to the
probability coefficient for 𝑐𝑖 = 3 in 𝐿 2(𝑥𝑖 ,𝑦𝑖)
By imposing the listed assumptions, I obtain the convex combination coefficients
matrix listed in Figure 1.
Figure 1: Hand-type and lottery-type convex combination coefficients matrix
Player I’s Hand-Type
𝒄𝒊 = 𝟏 𝒄𝒊 = 𝟐 𝒄𝒊 = 𝟑
Player I’s
Lottery-Type
𝑳𝟏(𝒙𝒊,𝒚𝒊) 𝑦𝑖 1
2(1 − 𝑥𝑖) 1 −
1
2(1 − 𝑥𝑖) − 𝑦𝑖
𝑳𝟐(𝒙𝒊,𝒚𝒊) 1
2(1 − 𝑥𝑖) 𝑥𝑖
1
2(1 − 𝑥𝑖)
𝑳𝟑(𝒙𝒊,𝒚𝒊) 1 −1
2(1 − 𝑥𝑖) − 𝑦𝑖
1
2(1 − 𝑥𝑖) 𝑦𝑖
Since the coefficients of a convex combination must be non-negative and sum
to 1, I have included the Boundary constraints 𝑥𝑖 ∈ [0, 1] and 𝑦𝑖 ∈ [0,1], along with
the Negativity constraint 1
2(1 − 𝑥𝑖) + 𝑦𝑖 ≤ 1. The Boundary constraints ensure that
the parameters on the main diagonal of the convex combination coefficients matrix
remain non-negative with an upper bound of 1, while the Negativity constraint
guarantees that all other parameters of this matrix are also non-negative with an
upper bound of 1. Finally, I have imposed the Expected Value constraint 4𝑦𝑖 − 𝑥𝑖 ≥ 1,
which is a structural assumption that assures that 𝐸[𝐿 3(𝑥𝑖 ,𝑦𝑖)] ≥ 𝐸[𝐿2(𝑥𝑖 ,𝑦𝑖)] ≥
𝐸[𝐿1(𝑥𝑖 ,𝑦𝑖)]. By applying this condition, and assuming the risk-neutrality of players,
I am able to create an ordinal ranking for each player’s preferences toward lottery-
types, such that 𝐿 3(𝑥𝑖 ,𝑦𝑖) ≿ 𝐿2(𝑥𝑖 ,𝑦𝑖) ≿ 𝐿 1(𝑥𝑖 ,𝑦𝑖). Upon considering each of the listed
constraints, the shaded region in Figure 2 shows the valid informational ability
coordinates for each player in my model.
Figure 2:
0
0.05
0.10.15
0.2
0.250.3
0.35
0.40.45
0.5
0.55
0.60.65
0.7
0.750.8
0.85
0.90.95
1
0
0.0
3
0.0
6
0.0
9
0.1
2
0.1
5
0.1
8
0.2
1
0.2
4
0.2
7
0.3
0.3
3
0.3
6
0.3
9
0.4
2
0.4
5
0.4
8
0.5
1
0.5
4
0.5
7
0.6
0.6
3
0.6
6
0.6
9
0.7
2
0.7
5
0.7
8
0.8
1
0.8
4
0.8
7
0.9
0.9
3
0.9
6
0.9
9
Y -
([1
] &
[3
] Lo
tter
y V
alu
es)
X - ([2] Lottery Value)
Range of Valid Informational Ability Coordinates
Expected Value Constraint Negativity Constraint
3.2.2 Perceptive Skill
According to Caro (2003), “once you’ve mastered the basic elements of a
winning poker formula, psychology becomes the key ingredient in separating break-
even players from players who win consistently. The most profitable kind of poker
psychology is the ability to read your opponents.” With this in mind, I implemented
a perceptive ability, to coincide with the technical skill, for each player. The
perceptive ability is meant to replicate a poker player’s knack for “reading” their
opponent.
In my model, the perceptive ability (𝑝𝑖) endowed to player 𝑖 is taken from the
set {1, 4}, and determines the specific number of informational ability coordinates that
player 𝑖 believes that player −𝑖 may have. For example, suppose 𝑝𝑖 = 4. Player 𝑖 will
then believe that player −𝑖 is endowed with one of four, possible, unique
informational ability coordinates, where one of which is actually correct. Additionally,
I assume that player 𝑖 also believes that there is an equal probability of each of these
informational ability coordinates being player −𝑖’s true coordinate. For instance,
suppose 𝑝𝑖 = 4 and (𝑥−𝑖 , 𝑦−𝑖) = (3
4,
3
4). Player 𝑖 will believe that there is a 25% chance
that player −𝑖 has the informational ability coordinate (𝑥−𝑖 ,𝑦−𝑖) = (3
4,
3
4), and a 25%
chance that player −𝑖 has another, unique, coordinate for three additional
informational ability coordinates6.
6 These additional informational ability coordinates will be chosen exogenously.
Upon considering the above example, it is clear that if 𝑝𝑖 = 1 then player 𝑖 has
perfect information regarding player −𝑖’s technical skill. Conversely, if 𝑝𝑖 = 4 then
player 𝑖 has imperfect information regarding player −𝑖’s technical skill. Finally, it is
important to note that, in my model, I am assuming that both players have perfect
information regarding the other player’s perceptive ability. That is, if 𝑝𝑖 = 𝑧; (𝑖 ∈ {1, 2}
and 𝑧 ∈ {1, 4}), then player −𝑖 knows that 𝑝𝑖 = 𝑧, and vice versa for player −𝑖.
3.2.3 Luck
The final feature of my model that I will discuss is the luck component that I
have incorporated. As quoted in Rodman et al. (2009), “to win tournaments, you’ve
got to play well and you’ve got to get lucky in some key spots”. Luck, in my model,
arises when both players specifically choose “Call”. When this happens, a showdown
occurs, and additional values, 𝛼𝑖 , 𝛽𝑖 and 𝛾𝑖 , are drawn for player 𝑖 (𝑖 ∈ {1,2}). Since
𝛼𝑖 , 𝛽𝑖 , and 𝛾𝑖 are drawn from the same distribution as player 𝑖’s hand-type, 𝑐𝑖, and
since all four parameters carry an equal weight when determining the updated
capital values, each player, in this case, will have made the decision to “Call” with
information that pertains to only one quarter of their total hand value (henceforth
referred to as a player’s score). So, regardless of what hand-type, 𝑐𝑖, players are dealt,
either player still has a chance to win. For instance, consider the case when player 𝑖
is dealt 𝑐𝑖 = 1, and player −𝑖 is dealt 𝑐−𝑖 = 3. If the additional values for player −𝑖 are
𝛼−𝑖 = 1, 𝛽−𝑖 = 1, and 𝛾−𝑖 = 1, then player −𝑖’s score would be 𝑐−𝑖 + 𝜆−𝑖 = 𝑐−𝑖 + 𝛼−𝑖 +
𝛽−𝑖 + 𝛾−𝑖 = 6. Now, since 𝑐𝑖 = 1, any combination of 𝛼𝑖, 𝛽𝑖, and 𝛾𝑖 such that 𝛼𝑖 + 𝛽𝑖 +
𝛾𝑖 ≥ 6 = 𝑐−𝑖 + 𝜆−𝑖 would lead to player 𝑖 winning the showdown over player −𝑖, despite
player 𝑖 drawing the lowest, initial, hand-type and player −𝑖 drawing the highest,
initial, hand-type. Appendix I lists the probabilities of a player winning, drawing, or
losing a showdown for every initial combination of hand-types.
4 Calibration
Prior to conducting a comparative static analysis on the model, I would first
like to calibrate some parameters to depict a benchmark case that replicates a
common situation for a poker hand. My reasoning for doing so will be outlined in the
following paragraphs. A summary of the benchmark structural parameters I utilize
is listed in Figure 3.
Figure 3: Structural parameters used in the Benchmark Case78
Parameter Value Interpretation
𝑘10 10 Player 1’s initial capital level
𝑘20 10 Player 2’s initial capital level
𝑏 1 The blind cost
(𝑥1,𝑦1) (1, 1) Player 1’s technical skill
(𝑥2,𝑦2 ) (1, 1) Player 2’s technical skill
𝑝1 1 Player 1’s perceptive skill
𝑝2 1 Player 2’s perceptive skill
At the beginning of any standard poker tournament, all players have the same
amount of capital. So, for the benchmark case, I chose to set the initial capital level
for player 1 equal to the initial capital level for player 2. In some tournaments,
however, players may start with varying levels of capital. Also, as a tournament
7 For the benchmark case, I have set (𝑥1 ,𝑦1
) = (𝑥2 ,𝑦2) = (1, 1) to give a baseline assumption that each player has
perfect technical skill . 8 For the benchmark case, I have set 𝑝1 = 𝑝2 = 1 to give a baseline assumption that each player has perfect
perceptive skil l.
progresses, players will very likely have disparate capital levels. Accordingly, I will
conduct analysis based on how player behaviour changes as I alter the capital-level
ratio between the two players. Nevertheless, for the benchmark case, I have assumed
capital-level symmetry between the two players.
Given that 𝑘10 = 𝑘2
0 = 𝑘0, an interesting property of the model is that the
absolute capital and blind levels do not, necessarily, matter for the purpose of
analysis. Instead, it is the relative values between these parameters that are
important. Since I have assumed risk-neutrality for both players, the payoffs for each
player will always be scaled by the same factor as that of which the structural
parameters are scaled by. For example, consider the case where I scale the
parameters by a factor of 𝑥. This will result with the parameters being 𝑘1∗ = 𝑘2
∗ = 𝑥𝑘0
and 𝑏∗ = 𝑥𝑏. The payoffs, for each player (𝑖 ∈ {1, 2}), will be as follows:
If both players were to “Fold”
𝑈𝑖( ∙ ) = 𝑘𝑖 = 𝑘𝑖∗ − 𝑏∗ + 𝑏∗ = 𝑥𝑘𝑖
0 (7)
If player 𝑖 were to “Call” and player −𝑖 were to “Fold”:
𝑈𝑖( ∙ ) = 𝑘𝑖 = 𝑘𝑖∗ + 𝑏∗ = 𝑥𝑘𝑖
0 + 𝑥𝑏 = 𝑥(𝑘𝑖0 + 𝑏) (8)
𝑈−𝑖( ∙ ) = 𝑘−𝑖 = 𝑘−𝑖∗ − 𝑏∗ = 𝑥𝑘−𝑖
0 − 𝑥𝑏 = 𝑥(𝑘−𝑖0 − 𝑏) (9)
If player 𝑖 were to “Fold” and player −𝑖 were to “Call”:
𝑈𝑖( ∙ ) = 𝑘𝑖 = 𝑘𝑖∗ − 𝑏∗ = 𝑥𝑘𝑖
0 − 𝑥𝑏 = 𝑥(𝑘𝑖0 − 𝑏) (10)
𝑈−𝑖( ∙ ) = 𝑘−𝑖 = 𝑘−𝑖∗ + 𝑏∗ = 𝑥𝑘−𝑖
0 + 𝑥𝑏 = 𝑥(𝑘−𝑖0 + 𝑏) (11)
If both players were to “Call”:
𝑈𝑖( ∙ ) = 𝑘𝑖
= {
𝑘𝑖∗ + 𝑀𝑖𝑛{𝑘𝑖
∗, 𝑘−𝑖∗ } = 𝑥(𝑘𝑖
0 + 𝑀𝑖𝑛{𝑘𝑖0, 𝑘−𝑖
0 }) 𝑖𝑓 𝑐𝑖 + 𝜆𝑖 > 𝑐−𝑖 + 𝜆−𝑖
𝑘𝑖∗ − 𝑀𝑖𝑛{𝑘𝑖
∗, 𝑘−𝑖∗ } = 𝑥(𝑘𝑖
0 − 𝑀𝑖𝑛{𝑘𝑖0, 𝑘−𝑖
0 }) 𝑖𝑓 𝑐𝑖 + 𝜆𝑖 < 𝑐−𝑖 + 𝜆−𝑖
𝑘𝑖∗ = 𝑥𝑘𝑖
0 𝑖𝑓 𝑐𝑖 + 𝜆𝑖 = 𝑐−𝑖 + 𝜆−𝑖
(12)
𝑈−𝑖( ∙ ) = 𝑘−𝑖
= {
𝑘−𝑖∗ + 𝑀𝑖𝑛{𝑘𝑖
∗, 𝑘−𝑖∗ } = 𝑥(𝑘−𝑖
0 + 𝑀𝑖𝑛{𝑘𝑖0, 𝑘−𝑖
0 }) 𝑖𝑓 𝑐−𝑖 + 𝜆−𝑖 > 𝑐𝑖 + 𝜆𝑖
𝑘−𝑖∗ − 𝑀𝑖𝑛{𝑘𝑖
∗, 𝑘−𝑖∗ } = 𝑥(𝑘−𝑖
0 − 𝑀𝑖𝑛{𝑘𝑖0, 𝑘−𝑖
0 }) 𝑖𝑓 𝑐−𝑖 + 𝜆−𝑖 < 𝑐𝑖 + 𝜆𝑖
𝑘−𝑖∗ = 𝑥𝑘−𝑖
0 𝑖𝑓 𝑐−𝑖 + 𝜆−𝑖 = 𝑐𝑖 + 𝜆𝑖
(13)
Therefore, based on the above results, if 𝑘10 = 𝑘2
0, then it is not the absolute values of
𝑘10, 𝑘2
0, and 𝑏 that matter for analysis, but instead it is the relative values of these
parameters that we should focus on.
A common inflection point for players in a poker tournament occurs when a
player’s capital level reaches a total of ten big blinds9. At this point, it is widely
known, amongst experienced players, that the benefits of aggressive play rise relative
to the benefits of passivity. Furthermore, at this stage, common advice to players is
to employ a strategic action set that consists of only two moves: raise all of your chips
or fold (Snyder, 2006). This advice was a contributing factor to my choice of utilizing
{𝐹𝑜𝑙𝑑, 𝐶𝑎𝑙𝑙} as the players’ action set; where, in my model, call is synonymous to raise
all of your chips. By setting 𝑘10 = 𝑘2
0 = 10 and 𝑏 = 1, I have effectively imposed, on
each player, a capital-to-blind ratio of 10-to-1, thereby making the action set
{𝐹𝑜𝑙𝑑, 𝐶𝑎𝑙𝑙} realistic. A noteworthy point to make is that Snyder (2006) advises
players with more than ten big blinds to also consider the action of raising to a value
that is less than all of your chips. In the Analysis section, I will alter the capital-to-
blind ratio of each player to determine optimal player behaviour when this ratio
exceeds 10-to-1 and when it is strictly less than 10-to-1.
9 Please see “Texas Hold’em Rules” (2016) for an explanation of what a big blind is specifically.
For convenience, I limited the set of possible informational ability coordinates,
for each player, to four. A summary of these four coordinates is listed in Figure 4. My
methodology for choosing the four coordinates is simple. First, I restricted the set of
possible coordinates to values of 𝑥𝑖 and 𝑦𝑖 (𝑖 ∈ {1,2}) that satisfy the Boundary,
Negativity, and Expected Value constraints, and are such that 𝑥𝑖 = 𝑦𝑖. Next, I
identified the coordinate that corresponds to perfect informational ability10, (𝑥𝑖 ,𝑦𝑖) =
(1,1), and the coordinate that corresponds to completely imperfect informational
ability11, (𝑥𝑖 ,𝑦𝑖) = (13⁄ , 1
3⁄ ). I, then, computed the expected values for each lottery-
type, for the perfect informational ability and completely imperfect informational
ability cases. Next, I took the difference between the expected value of 𝐿 3(1,1) and the
expected value of 𝐿1(1,1). I found this difference to equal 2. I repeated this for the
completely imperfect informational ability case, and found this expected value
difference to equal 0. Using these two extremes, I partitioned the range of expected
value differences into three equivalent regions, which provided me with two more
expected value differences to target, 23⁄ and 4
3⁄ . Finally, I solved for the
informational ability coordinates, (59⁄ , 5
9⁄ ) and (79⁄ , 7
9⁄ ), which respectively yielded
the two desired expected value differences. I subsequently used these two coordinates
as the middling, possible, informational ability coordinates for each player.
10 When a player has perfect informational ability, they are able to completely discern their hand type based on the specific lottery-type they draw. See Appendix A2 for the corresponding convex combination coefficients matrix for the case where (𝑥 𝑖 ,𝑦𝑖
) = (1,1). 11 When a player has completely imperfect informational ability, they are unable to attain, from the lottery-type they draw, any additional knowledge regarding their hand-type. See Appendix A2 for the corresponding convex
combination coefficients matrix for the case where (𝑥 𝑖 ,𝑦𝑖) = (1
3⁄ , 13⁄ ).
Furthermore, it is trivial, but important, to note that player 𝑖’s (𝑖 ∈ {1,2}) ability to
discern their true hand-type is strictly increasing as 𝑥𝑖 = 𝑦𝑖 increases from 13⁄ to 1.
Therefore, we can make the following general statement regarding player 𝑖’s
preferences over their informational ability coordinate (𝑥𝑖 ,𝑦𝑖):
(1, 1) ≿ (79⁄ , 7
9⁄ ) ≿ (59⁄ , 5
9⁄ ) ≿ (13⁄ , 1
3⁄ ) (14)
Additionally, for each chosen informational ability coordinate, the resulting
lottery-type convex combination coefficients matrix is displayed in Appendix A2.
Finally, in Appendix A3, I have included a plot that exhibits the chosen informational
ability coordinates, along with the Boundary, Negativity and Expected Value
constraints that I have imposed.
Figure 4: Informational ability coordinates used for analysis (for 𝑖 ∈ {1,2})
Parameter Value Interpretation Terminology EV Difference
(𝑥𝑖 ,𝑦𝑖)
(13⁄ , 1
3⁄ )
Player 𝑖 is completely
unable to interpret their
hand-type
“Completely Imperfect”
informational ability
0
(59⁄ , 5
9⁄ )
Player 𝑖 has a vague
ability to accurately
interpret their hand-type
“Vague” informational
ability
2
3⁄
(79⁄ , 7
9⁄ )
Player 𝑖 has a strong
ability to accurately
interpret their hand-type
“Strong” informational
ability
4
3⁄
(1, 1)
Player 𝑖 is able to
perfectly interpret their
hand-type
“Perfect” informational
ability
2
5 Analysis
In this section, I will present a comparative static analysis based on altering
the parameters of my model from the parameters I outlined for the benchmark case.
A summary of the cases presented in this section is listed in Appendix A4. Before
proceeding, I would like to detail some preliminary insights that we should consider
throughout our analysis.
5.1 Preliminaries
The solution concept I use is that of a Bayesian-Nash equilibrium12. Recall that
a Bayesian-Nash equilibrium, for a two-player game, is a strategy profile and belief
specification, for each player regarding the types of the other player, that maximizes
the expected payoff for each player given their beliefs about the other player’s type(s)
and given the other player’s strategy. Also, recall that a strategy for a Bayesian game
is a complete, contingent plan that specifies a player’s action for every type that
player may be. Now, relating this to my model, notice that player 𝑖’s (𝑖 ∈ {1, 2})
strategy (𝑠𝑖) will be a vector consisting of the following three parameters:
𝑎𝑖(𝐿1(𝑥𝑖 ,𝑦𝑖)), 𝑎𝑖(𝐿2(𝑥𝑖 ,𝑦𝑖)), and 𝑎𝑖(𝐿3(𝑥𝑖 ,𝑦𝑖)). The notation I use to denote player 𝑖’s
strategy is:
𝑠𝑖 = (𝑎𝑖(𝐿1(𝑥𝑖 ,𝑦𝑖)),𝑎𝑖(𝐿 2(𝑥𝑖 ,𝑦𝑖)),𝑎𝑖(𝐿3(𝑥𝑖 ,𝑦𝑖))) (15)
12 The format I use to display the Bayesian-Nash equilibrium is {𝑠1 , 𝑠2}, where 𝑠𝑖 (𝑖 ∈ {1,2}) represents the Bayesian-
Nash equilibrium strategy vector for player 𝑖.
The reason that player 𝑖’s strategy only contains three parameters, despite the many
types that player 𝑖 may be, is that many of the types, specifically player 𝑖’s
informational ability and perceptive ability, will be predefined for each case that I
conduct. Furthermore, notice that I have also not included player 𝑖’s hand-type in
their strategy vector. My reason for this is that player 𝑖’s hand-type is essentially
drawn after player 𝑖 has already chosen their strategy. Player 𝑖’s specific hand-type
only becomes a factor when computing player 𝑖’s score in a showdown, whereas player
𝑖’s strategy is selected based on an expectation of the hand-type they will draw.
Additionally, since both players are risk-neutral and receive payoffs equal to
their post-game capital level, each case I analyze will be a zero-sum game. Therefore,
the sum of the payoffs the players receive will always equal the sum of the players’
initial capital levels.
Finally, I would like to summarize the beliefs that player 𝑖 (𝑖 ∈ {1, 2}) has in
the game I outlined above, given player −𝑖’s perceptive ability, 𝑝−𝑖, as well as both
players’ informational ability, (𝑥𝑖 ,𝑦𝑖) and (𝑥−𝑖 ,𝑦−𝑖). First, player 𝑖 believes that, given
the lottery-type, 𝐿𝑗(𝑥𝑖 ,𝑦𝑖) (𝑗 ∈ {1, 2, 3}), the probability that player 𝑖 has of drawing a
specific hand-type, 𝑐𝑖, is equal to the corresponding value of the convex combination
coefficients matrix shown in Figure 1 (on page 11). Second, player 𝑖 believes that
player −𝑖 has a 1
3 probability of drawing each specific hand-type, 𝑐−𝑖, and each specific
lottery-type, 𝐿𝑗(𝑥−𝑖 ,𝑦−𝑖). Third, if 𝑝𝑖 = 1, then player 𝑖 believes for certain that player
−𝑖 has the informational ability, (𝑥−𝑖 , 𝑦−𝑖). Fourth, if 𝑝𝑖 = 4, then player 𝑖 believes that
there is a 1
4 probability that player −𝑖 has the informational ability, (𝑥−𝑖 , 𝑦−𝑖), as well
as a 1
4 probability that player −𝑖 has each of the remaining, possible, informational
abilities (�̇�−𝑖 , �̇�−𝑖), (�̈�−𝑖 , �̈�−𝑖), and (𝑥−𝑖 , 𝑦−𝑖)13. Finally, player 𝑖 believes for certain that
player −𝑖 has the perceptive ability, 𝑝−𝑖.
5.2 Case Analysis
In this subsection, I will conduct a case-by-case comparative static analysis on
my model. I will begin each case by presenting an overview of the structural
parameters I used and the results I obtained for that specific case. I will follow each
case with a brief discussion detailing the corresponding results. Although most of the
results are easy to interpret, it is essential for me to elaborate on the difference
between player 𝑖’s (𝑖 ∈ {1, 2}) expected payoff and player 𝑖’s actual payoff. Player 𝑖’s
expected payoff is the payoff that player 𝑖 expects to attain after considering all of
their beliefs and choosing an action, but before player −𝑖’s perceptive ability is
revealed to them. Player 𝑖’s actual payoff is the payoff that player 𝑖 would expect to
receive after considering all of their beliefs, choosing an action, and observing player
−𝑖’s perceptive ability.
13 That is, if 𝑝𝑖 = 4, then player 𝑖 believes that there is a
1
4 probability that player −𝑖 has each of the following
informational abilities: (1,1), (79⁄ , 7
9⁄ ), (59⁄ , 5
9⁄ ), (13⁄ , 1
3⁄ ).
5.2.1 Benchmark Case
Figure 5: Structural parameters & results for the Benchmark Case
Sub-case
Player Blind Cost
Initial Capital
Technical Skill
Perceptive Skill14
Bayesian-Nash
Equilibria
Expected Payoff
Actual Payoff
BM 1
1 10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
The benchmark case is quite straightforward, with only a few noteworthy
results. The first is that, since this game is symmetric, the actual payoffs for each
player, 𝑖 (𝑖 ∈ {1,2}), will be equivalent to that of the other player, −𝑖. Furthermore,
this implies that, since this is a two-player, zero-sum game, the actual payoffs for
each player will equal the players’ initial capital level. Additionally, since both
players have perfect perceptive skill and the same level of technical skill, neither
player will expect to have an advantage over their opponent, thereby resulting in the
players’ expected payoffs to be equal to the players’ initial capital level.
Finally, notice that the Bayesian-Nash equilibrium for this case is
{(𝐹, 𝐹, 𝐶),(𝐹, 𝐹, 𝐶)}. Therefore, in this equilibrium, both players will be playing a
conservative, yet selectively aggressive strategy where each player will call with
𝐿 3(1,1), but fold with 𝐿 2(1,1) and 𝐿1(1,1). This will be important to consider when
comparing the Bayesian-Nash equilibria that arise from the other cases I study.
14 Under Perceptive Skill, “Perfect” corresponds to a player, 𝑖 (𝑖 ∈ {1,2}), such that 𝑝𝑖 = 1; “Imperfect” corresponds
to a player, 𝑖 (𝑖 ∈ {1,2}), such that 𝑝𝑖 = 4.
5.2.2 Altering the Blind Cost
Figure 6: Results obtained when altering the blind cost (𝑏):
Sub-case:
Player Blind Cost
Initial Capital
Technical Skill
Perceptive Skill
Bayesian-Nash
Equilibria
Expected Payoff
Actual Payoff
1 1
0 10 Perfect Perfect
Multiple15 10 10
2 10 Perfect Perfect 10 10
2 1
0.01 10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
3 1
0.5 10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
4 (BM)
1 1
10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
5 1 889
729⁄ 10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
6 1 890
729⁄ 10 Perfect Perfect
Multiple16 10 10
2 10 Perfect Perfect 10 10
7 1 891
729⁄ 10 Perfect Perfect (F,C,C) 10 10
2 10 Perfect Perfect (F,C,C) 10 10
8 1
2 10 Perfect Perfect (F,C,C) 10 10
2 10 Perfect Perfect (F,C,C) 10 10
9 1
3 10 Perfect Perfect (F,C,C) 10 10
2 10 Perfect Perfect (F,C,C) 10 10
10 1
4 10 Perfect Perfect (C,C,C) 10 10
2 10 Perfect Perfect (C,C,C) 10 10
11 1
5 10 Perfect Perfect (C,C,C) 10 10
2 10 Perfect Perfect (C,C,C) 10 10
For this study, I held each of the structural parameters, except for the blind
cost (𝑏), consistent to that of the benchmark case. By altering the value of the blind
15 The following four Bayesian-Nash equilibria were found for this case: {(𝐹, 𝐹, 𝐹), (𝐹, 𝐹, 𝐹)}, {(𝐹 , 𝐹, 𝐹), (𝐹 ,𝐹, 𝐶)}, {(𝐹, 𝐹, 𝐶), (𝐹, 𝐹, 𝐹)}, and {(𝐹, 𝐹, 𝐶), (𝐹, 𝐹, 𝐶)}. Each equilibrium resulted in both players receivi ng an expected
payoff of 10 and an actual payoff of 10. 16 The following three Bayesian-Nash equilibria were found for this case: {(𝐹, 𝐹, 𝐶), (𝐹, 𝐹, 𝐶)}, {(𝐹 , 𝐹, 𝐶), (𝐹, 𝐶, 𝐶)},
and {(𝐹, 𝐶, 𝐶), (𝐹, 𝐹, 𝐶)}. Each equilibrium resulted in both players receivi ng an expected payoff of 10 and an actual
payoff of 10.
cost, I am essentially changing the expense of folding as well as the additional payoff
a player would receive by calling when their opponent folds.
First, for the same reasons outlined in the benchmark case, the expected and
actual payoffs for each player, in this situation, will equal the players’ initial capital
level. Second, an interesting characteristic of this case is that as the blind cost
increases, the players’ Bayesian-Nash equilibria strategies become more aggressive.
This result is intuitive because as the blind cost increases, both the price of folding
and the potential reward for calling increases, which thereby magnifies the benefits
of aggressive play. This result is also consistent with common poker insight that
advises poker players to play more aggressively as the blinds increase.
Finally, it is interesting to note the subcases where multiple Bayesian-Nash
equilibria exist. The blind costs for these cases are threshold points where the
equilibrium strategies switch from a more conservative strategy to a more aggressive
strategy. Subcase 1 is particularly interesting because of the existence of Bayesian-
Nash equilibrium strategies that have players of perfect technical skill folding the
highest lottery-type, which is, essentially, the best possible hand-type since both
players have perfect informational ability. However, this is not difficult to reconcile
since a blind cost of zero implies that the players will not incur a cost of folding, and
since both players know that the other player is of the same technical ability, neither
will have an advantage of playing, which therefore makes folding all hands a viable
strategy.
5.2.3 Altering the Players’ Initial Capital Ratio
Figure 7: Results obtained when altering the players’ initial capital ratio (𝑘10
𝑘20⁄ ):
Sub-case
Player Blind Cost
Initial Capital
Technical Skill
Perceptive Skill
Bayesian-Nash
Equilibria
Expected Payoff
Actual Payoff
1 1
1 1 Perfect Perfect (C,C,C) 1 1
2 10 Perfect Perfect (C,C,C) 10 10
2 1
1 2 Perfect Perfect (C,C,C) 2 2
2 10 Perfect Perfect (C,C,C) 10 10
3 1
1 2.915 Perfect Perfect (C,C,C) 2.915 2.915
2 10 Perfect Perfect (C,C,C) 10 10
4 1
1 2.916 Perfect Perfect
Multiple17 2.916 2.916
2 10 Perfect Perfect 10 10
5 1
1 2.917 Perfect Perfect (F,C,C) 2.917 2.917
2 10 Perfect Perfect (F,C,C) 10 10
6 1
1 5 Perfect Perfect (F,C,C) 5 5
2 10 Perfect Perfect (F,C,C) 10 10
7 1
1 8.19 Perfect Perfect (F,C,C) 8.19 8.19
2 10 Perfect Perfect (F,C,C) 10 10
8 1
1 7290
890⁄ Perfect Perfect Multiple18
7290890⁄ 7290
890⁄
2 10 Perfect Perfect 10 10
9 1
1 8.2 Perfect Perfect (F,F,C) 8.2 8.2
2 10 Perfect Perfect (F,F,C) 10 10
10 (BM)
1 1
10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
11 1
1 100 Perfect Perfect (F,F,C) 100 100
2 10 Perfect Perfect (F,F,C) 10 10
17 The following four Bayesian-Nash equilibria were found for this case: {(𝐹 , 𝐶, 𝐶), (𝐹, 𝐶, 𝐶)}, {(𝐹, 𝐶, 𝐶), (𝐶, 𝐶, 𝐶)}, {(𝐶, 𝐶, 𝐶), (𝐹, 𝐶, 𝐶)}, and {(𝐶, 𝐶, 𝐶), (𝐶, 𝐶, 𝐶)}. Each equilibrium resulted in players 1 and 2 receiving expected payoffs of 2.916 and 10 respectively, and actual payoffs of 2.916 and 10 respectively. 18 The following three Bayesian-Nash equilibria were found for this case: {(𝐹, 𝐹, 𝐶), (𝐹, 𝐹, 𝐶)}, {(𝐹 , 𝐹, 𝐶), (𝐹, 𝐶, 𝐶)},
and {(𝐹, 𝐶, 𝐶), (𝐹, 𝐹, 𝐶)}. Each equilibrium resulted in players 1 and 2 receiving expected payoffs of 7290890⁄ and
10 respectively, and actual payoffs of 7290890⁄ and 10 respectively.
For this case, I held the structural parameters consistent to that of the
benchmark case, except, this time, I altered the players’ initial capital ratio, (𝑘10
𝑘20⁄ ).
By changing this ratio, I effectively made one player, initially, richer in capital than
the other.
For similar reasons as in the previous cases, the expected and actual payoffs
for each player will equal the players’ initial capital level. Also, notice that as I
increase the players’ initial capital ratio, the Bayesian-Nash equilibrium strategies
for both players become more conservative. It makes sense that player 1 would want
to play an aggressive strategy since the blind cost is proportionally higher to their
initial capital level when their initial capital level is low. It’s somewhat unintuitive
as to why player 2’s equilibrium strategy always seems to mirror player 1’s
equilibrium strategy. However, the reason for this is that when 𝑘20 > 𝑘1
0, player 2 can
only lose a maximum of 𝑘10, and since both players are risk-neutral, each player will
value the risk equal to that of the other player. This, thereby, induces strategic
symmetry between player 1 and player 2, despite their different initial capital levels.
Furthermore, for this case, when 𝑘𝑖0 > 𝑘−𝑖
0 (𝑖 ∈ {1,2}), the game presented is
equivalent to that of where 𝑘𝑖0 = 𝑘−𝑖
0 , and player 𝑖 has 𝑘𝑖0 − 𝑘−𝑖
0 added to their post-
game capital level. Now, recall our conjecture in the Calibration section, such that it
is not the absolute values of 𝑘𝑖0, 𝑘−𝑖
0 , and 𝑏 that matter for analysis, but instead it is
the relative values of these parameters that are important. With this in mind, the
deterministic parameters in this case are �̂� = 𝑀𝑖𝑛{𝑘10, 𝑘2
0} and 𝑏, which effectively
makes it identical to the case where I only altered the blind cost. This can be verified
by comparing the subcases, for each case, that have identical �̂�-to-𝑏 ratios19.
5.2.4 Altering the Players’ Skill Level
Figure 8: Results obtained when altering the players’ technical skill (𝑥𝑖 ,𝑦𝑖) & perceptive skill (𝑝𝑖):
Sub- case:
Player Blind Cost
Initial Capital
Technical Skill
Perceptive Skill
Bayesian-Nash Equilibria
Expected Payoff
Actual Payoff
1 (BM)
1 1
10 Perfect Perfect (F,F,C) 10 10
2 10 Perfect Perfect (F,F,C) 10 10
2 1
1 10 Perfect Perfect (F,F,C) 10.0736 10.0736
2 10 Perfect Imperfect (F,C,C) 10.3204 9.9264
3 1
1 10 Perfect Imperfect (F,C,C) 10.381 10
2 10 Perfect Imperfect (F,C,C) 10.381 10
4 1
1 10 Perfect Perfect (F,C,C) 10.189 10.189
2 10 Strong Perfect (F,F,C) 9.811 9.811
5 1
1 10 Perfect Perfect (F,C,C) 10.254 10.254
2 10 Strong Imperfect (F,C,C) 10.127 9.746
6 1
1 10 Perfect Imperfect (F,C,C) 10.3204 10.189
2 10 Strong Perfect (F,F,C) 9.811 9.811
7 1
1 10 Perfect Imperfect (F,C,C) 10.381 10.254
2 10 Strong Imperfect (F,C,C) 10.127 9.746
8 1
1 10 Perfect Perfect (F,C,C) 10.4517 10.4517
2 10 Vague Perfect (F,F,C) 9.5483 9.5483
9 1
1 10 Perfect Perfect (F,C,C) 10.5081 10.5081
2 10 Vague Imperfect (F,C,C) 9.873 9.4919
10 1
1 10 Perfect Imperfect (F,C,C) 10.3204 10.4517
2 10 Vague Perfect (F,F,C) 9.5483 9.5483
11 1
1 10 Perfect Imperfect (F,C,C) 10.381 10.5081
2 10 Vague Imperfect (F,C,C) 9.873 9.4919
12 1
1 10 Perfect Perfect No Pure Strategy Bayesian-Nash
Equilibria20 2 10 C. Imperfect Perfect
13 1
1 10 Perfect Perfect (F,C,C) 10.8098 10.8098
2 10 C. Imperfect Imperfect (C,C,C) 9.7618 9.1902
14 1
1 10 Perfect Imperfect No Pure Strategy Bayesian-Nash
Equilibria20 2 10 C. Imperfect Perfect
19 Specifically notice that the Bayesian-Nash equilibria results and 𝑘-to-𝑏 ratios are equivalent in the following
subcases: 5.2.2 (4) & 5.2.3 (10); 5.2.2 (6) & 5.2.3 (8); 5.2.2 (8) & 5.2.3 (6); and 5.2.2 (11) & 5.2.3 (2). 20 The following mixed strategy Bayesian-Nash equilibrium exists: {
50
57(𝐹, 𝐶, 𝐶) +
7
57(𝐶, 𝐶, 𝐶),
27
38(𝐹, 𝐹, 𝐹) +
11
38(𝐶, 𝐶, 𝐶)}. This equilibrium yields expected and actual payoffs of 10.71 and 9.29 for Player’s 1 and 2 respectively.
15 1
1 10 Perfect Imperfect (F,C,C) 10.8098 10.8098
2 10 C. Imperfect Imperfect (C,C,C) 9.7618 9.1902
16 1
1 10 Strong Perfect (F,C,C) 10 10
2 10 Strong Perfect (F,C,C) 10 10
17 1
1 10 Strong Perfect (F,C,C) 10 10
2 10 Strong Imperfect (F,C,C) 10.127 10
18 1
1 10 Strong Imperfect (F,C,C) 10.127 10
2 10 Strong Imperfect (F,C,C) 10.127 10
19 1
1 10 Strong Perfect (F,C,C) 10.254 10.254
2 10 Vague Perfect (F,C,C) 9.746 9.746
20 1
1 10 Strong Perfect (F,C,C) 10.254 10.254
2 10 Vague Imperfect (F,C,C) 9.873 9.746
21 1
1 10 Strong Imperfect (F,C,C) 10.127 10.254
2 10 Vague Perfect (F,C,C) 9.746 9.746
22 1
1 10 Strong Imperfect (F,C,C) 10.127 10.254
2 10 Vague Imperfect (F,C,C) 9.873 9.746
23 1
1 10 Strong Perfect (F,C,C) 10.4287 10.4287
2 10 C. Imperfect Perfect (C,C,C) 9.5713 9.5713
24 1
1 10 Strong Perfect (F,C,C) 10.4287 10.4287
2 10 C. Imperfect Imperfect (C,C,C) 9.7618 9.5713
25 1
1 10 Strong Imperfect (F,C,C) 10.4287 10.4287
2 10 C. Imperfect Perfect (C,C,C) 9.5713 9.5713
26 1
1 10 Strong Imperfect (F,C,C) 10.4287 10.4287
2 10 C. Imperfect Imperfect (C,C,C) 9.7618 9.5713
27 1
1 10 Vague Perfect (F,C,C) 10 10
2 10 Vague Perfect (F,C,C) 10 10
28 1
1 10 Vague Perfect (F,C,C) 10 10
2 10 Vague Imperfect (F,C,C) 9.873 10
29 1
1 10 Vague Imperfect (F,C,C) 9.873 10
2 10 Vague Imperfect (F,C,C) 9.873 10
30 1
1 10 Vague Perfect (F,C,C) 10.0477 10.0477
2 10 C. Imperfect Perfect (C,C,C) 9.9523 9.9523
31 1
1 10 Vague Perfect (F,C,C) 10.0477 10.0477
2 10 C. Imperfect Imperfect (C,C,C) 9.7618 9.9523
32 1
1 10 Vague Imperfect (F,C,C) 10.0477 10.0477
2 10 C. Imperfect Perfect (C,C,C) 9.9523 9.9523
33 1
1 10 Vague Imperfect (F,C,C) 10.0477 10.0477
2 10 C. Imperfect Imperfect (C,C,C) 9.7618 9.9523
34 1
1 10 C. Imperfect Perfect (C,C,C) 10 10
2 10 C. Imperfect Perfect (C,C,C) 10 10
35 1
1 10 C. Imperfect Perfect (C,C,C) 10 10
2 10 C. Imperfect Imperfect (C,C,C) 10 10
36 1
1 10 C. Imperfect Imperfect (C,C,C) 10 10
2 10 C. Imperfect Imperfect (C,C,C) 10 10
For this study, I held the blind cost and initial capital levels consistent to that
of the benchmark case, while testing every combination of technical and perceptive
skill between the players.
For subcases 2, 5, 6, 9, 13, 17, 20, and 24, when one player has perfect
perceptive skill while the other player does not, the player that does not begins the
game expecting to receive a certain payoff, but will actually yield a lesser payoff.
Whereas, the player with perfect perceptive ability will yield an equivalent payoff to
that which they expect. Subcase 2 is particularly interesting. Here, both players have
perfect technical skill, however, player 1 has perfect perceptive ability while player 2
has imperfect perceptive ability. Notice that if both players were able to discern the
other player’s technical ability, the Bayesian-Nash equilibrium would be
{(𝐹, 𝐹, 𝐶),(𝐹, 𝐹, 𝐶)}. Yet, since player 2 is unable to discern player 1’s technical skill,
the Bayesian-Nash equilibrium is, instead, {(𝐹, 𝐹, 𝐶),(𝐹, 𝐶, 𝐶)}. Therefore, in this
subcase, player 2 will be playing a non-optimal, more aggressive strategy than she
would have played if she had perfect perceptive ability. In addition to this, given each
player’s knowledge of their own perfect technical skill, both players expect to receive
a higher payoff than their initial capital level. Player 1, due to their perfect perceptive
ability, will accurately gauge their payoff, whereas player 2 will drastically
overestimate their payoff. The underlying reason behind player 2’s overestimation is
that player 2 believes that there is an equal probability of player 1 having a perfect,
strong, vague, or completely imperfect technical skill. Unfortunately for player 2,
player 1 has perfect technical skill, and is thereby able to capitalize on player 2’s
inability to discern player 1’s technical skill.
Now, consider subcase 10. The only structural difference between this subcase
and that of subcase 2 is that the player with perfect perceptive ability now has vague
technical skill. What’s interesting here is that player 1, who has perfect technical skill
but imperfect perceptive skill, expects the same payoff as they did in subcase 2.
However, this time, the payoff they actually receive will be higher than expected, for
a reason similar to that that was elaborated in the previous paragraph.
Another interesting result is that without the presence of a player with perfect
technical skill, altering the players’ perceptive ability, while holding all other
structural parameters constant, does not affect the Bayesian-Nash equilibria nor
does it affect the actual payoffs the players receive21. Therefore, for my case study,
the absence of a player with perfect technical skill implies that the perceptive abilities
only affect the payoff expectations that players may have. Additionally, there is no
case in which changing a player’s perceptive ability from perfect to imperfect, while
holding all else constant, benefits the player in terms of their actual payoff. However,
consider subcases 4 and 7. Here, it is interesting to note that if we take players with
perfect and strong technical skill respectively, and switch both of their perceptive
abilities from perfect to imperfect, the player with perfect technical skill will receive
21 This is true for the specific blind cost, initial capital levels, and technical skill possibilities that I have allowed for in this case. For example, changing the players’ perceptive abilities could also lead to different Bayesian-Nash equilibria outcomes and actual payoffs of the game in the absence of a player with perfect technical skill, (𝑥 𝑖 ,𝑦𝑖
) = (1, 1), but
in the presence of a player with almost perfect technical skil l (𝑖. 𝑒. (𝑥 𝑖 ,𝑦𝑖) = (0.99, 0.99)).
a higher actual payoff than before, whereas the player with strong technical skill will
receive a lower actual payoff than before. This also holds true if the player with strong
technical skill instead has vague or completely imperfect technical ability.
Next, notice that the Bayesian-Nash equilibria generated in this case indicate
that players with a high technical skill tend to play selectively aggressive strategies,
such as (𝐹, 𝐹, 𝐶) and (𝐹, 𝐶, 𝐶), in order to capitalize on the mistakes of lesser skilled
players22. Consider a player that has perfect technical skill. This player is able to
select a level of aggression that maximizes their payoff according to the technical skill
of their opponent. As shown in subcases 1, 4, 8, and 12, the perfectly skilled player is
able to adjust their level of aggression as the technical skill of their opponent
decreases. Furthermore, players with completely imperfect technical skill tend to
play recklessly aggressive strategies, such as (𝐶, 𝐶, 𝐶). This allows players with a high
technical skill to adjust their level of aggression in order to maximize their payoff.
These observations align with those stated by Rodman et al. (2009) and Siler (2010),
along with conventional poker wisdom that emphasizes the importance of adjusting
your strategy based on your opponents’ strategy.
Finally, to conclude the analysis, I will state some general observations that
we can learn from the results listed in Figure 8. Based on these subcases, it seems
that, in general, a player’s technical ability is the primary determinant of that
22 There are two types of mistakes that players can make in my model. These mistakes are:
I) Mistaking a poor hand for a good hand by drawing a high lottery-type and a low hand-type (for
example: A strong player that draws 𝐿3(79⁄ , 7
9⁄ ) and decides to call, but is actually dealt 𝑐𝑖 = 1).
II) Mistaking a good hand for a poor hand by drawing a low lottery-type and a high hand-type (for
example: A strong player that draws 𝐿1(79⁄ , 7
9⁄ ) and decides to fold, but is actually dealt 𝑐𝑖 = 3).
player’s payoff, whereas a player’s perceptive ability only serves to directly benefit
players of a high technical skill. While, for all other technical skill-types, a player’s
perceptive ability merely affects that player’s ability to gauge their actual payoff.
Also, subcase 2 generates an interesting result in the sense that the overconfidence
of a player with perfect technical skill and imperfect perceptive ability may get that
player into trouble when they enter the game expecting to win capital, but instead,
lose capital. This occurrence is similar to a realistic situation where a highly-skilled
online poker professional enters a poker game against a live-play poker professional.
Online poker players are known to have great technical ability, but sometimes lack
the perceptive skill that live players possess. In this case, the lack of perceptive skill
would lead to the live-play professional having an advantage over the online
professional. Furthermore, we should not discount the importance of a player’s
perceptive ability entirely. Notice that the perceptive skill is particularly important
when highly skilled players are present, which is true for most major poker
tournaments. This aligns with Caro’s (2003) claim that “once you’ve mastered the
basic elements of a winning poker formula, psychology becomes the key ingredient in
separating break-even players from players who win consistently. The most
profitable kind of poker psychology is the ability to read your opponents.”
6 Conclusion
The purpose of this paper was to present an original model, motivated by the
first betting round in a single, two-player, hand of No-Limit Hold’em poker. The
model I exhibit features two, risk-neutral, players engaging in a static, simultaneous-
move game of incomplete information, and incorporates luck as well as two different
types of skill, technical and perceptive. Through a comparative static analysis that
altered the structural parameters of my model, I developed insight regarding the
parametric effects of the model. Furthermore, the insights that I produce align with
conventional poker wisdom, as well as the results obtained by other works of
literature.
This paper is just the preliminary step of my research using this model, as
there are various extensions that I would like to make. First, I would like to link the
insights I obtained to online poker data. I plan to do this by using hand histories
compiled by HHSmithy.com, an online datamining service that tracks thousands of
online poker hands daily. Second, I would like to draw parallels between my model
and other strategic interactions in economics and industrial organization, then use
my model to develop insights on these other areas. Finally, I would like to incorporate
an experimental component that features subjects playing the game I outlined above.
It would be interesting to observe how experimental subjects approach my game, and
to see if the subjects act differently than my model predicts.
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8 Appendices
AI: Probability of each player winning, drawing, or losing a “showdown” for every initial
combination of hand-types:
Showdown Results - Probabilities
Player I's Hand-Type (Ci)
Player -I's Hand-Type (C-i)
Player I Wins & Player -I Loses
Players Draw Player I Loses & Player -I Wins
1 1 0.403 0.194 0.403
1 2 0.230 0.173 0.597
1 3 0.107 0.123 0.770
2 1 0.597 0.173 0.230
2 2 0.403 0.194 0.403
2 3 0.230 0.173 0.597
3 1 0.770 0.123 0.107
3 2 0.597 0.173 0.230
3 3 0.403 0.194 0.403
A2: Lottery-Type Convex Combination Coefficients Matrices:
“Completely Imperfect” Informational Ability ((𝑥𝑖 ,𝑦𝑖) = (13⁄ , 1
3⁄ )):
Player I’s Hand-Type
𝒄𝒊 = 𝟏 𝒄𝒊 = 𝟐 𝒄𝒊 = 𝟑 Expected Value
Player I’s
Lottery-
Type
𝑳𝟏(𝟏𝟑⁄ , 𝟏
𝟑⁄ ) 13⁄ 1
3⁄ 13⁄ 2
𝑳𝟐(𝟏𝟑⁄ , 𝟏
𝟑⁄ ) 13⁄ 1
3⁄ 13⁄ 2
𝑳𝟑(𝟏𝟑⁄ , 𝟏
𝟑⁄ ) 13⁄ 1
3⁄ 13⁄ 2
“Vague” Informational Ability ((𝑥𝑖 ,𝑦𝑖) = (59⁄ , 5
9⁄ )):
Player I’s Hand-Type
𝒄𝒊 = 𝟏 𝒄𝒊 = 𝟐 𝒄𝒊 = 𝟑 Expected Value
Player I’s
Lottery-
Type
𝑳𝟏(𝟓𝟗⁄ , 𝟓
𝟗⁄ ) 59⁄ 2
9⁄ 29⁄ 5
3⁄
𝑳𝟐(𝟓𝟗⁄ , 𝟓
𝟗⁄ ) 29⁄ 5
9⁄ 29⁄ 2
𝑳𝟑(𝟓𝟗⁄ , 𝟓
𝟗⁄ ) 29⁄ 2
9⁄ 59⁄ 7
3⁄
“Strong” Informational Ability ((𝑥𝑖 ,𝑦𝑖) = (79⁄ , 7
9⁄ )):
Player I’s Hand-Type
𝒄𝒊 = 𝟏 𝒄𝒊 = 𝟐 𝒄𝒊 = 𝟑 Expected Value
Player I’s
Lottery-
Type
𝑳𝟏(𝟕𝟗⁄ , 𝟕
𝟗⁄ ) 79⁄ 1
9⁄ 19⁄ 4
3⁄
𝑳𝟐(𝟕𝟗⁄ , 𝟕
𝟗⁄ ) 19⁄ 7
9⁄ 19⁄ 2
𝑳𝟑(𝟕𝟗⁄ , 𝟕
𝟗⁄ ) 19⁄ 1
9⁄ 79⁄ 8
3⁄
“Perfect” Informational Ability ((𝑥𝑖 ,𝑦𝑖) = (1, 1)):
Player I’s Hand-Type
𝒄𝒊 = 𝟏 𝒄𝒊 = 𝟐 𝒄𝒊 = 𝟑 Expected Value
Player I’s
Lottery-
Type
𝑳𝟏(𝟏,𝟏) 1 0 0 1
𝑳𝟐(𝟏,𝟏) 0 1 0 2
𝑳𝟑(𝟏,𝟏) 0 0 1 3
A3: Informational Ability Coordinates & Constraints – Calibration Plot:
A4: Summary of the cases presented in the Analysis section:
Section Case Number of Subcases
5.2.1 Benchmark 1
5.2.2 Altering the Blind Cost 11
5.2.3 Altering the Players' Initial Capital
Ratio 11
5.2.4 Altering the Players' Skill Level 36
(Xi , Yi ) = (1/3, 1/3)
(Xi , Yi ) = (5/9, 5/9)
(Xi , Yi ) = (7/9, 7/9)
(Xi , Yi ) = (1, 1)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Y -
([1
] &
[3
] Lo
tter
y V
alu
es)
X - ([2] Lottery Value)
Informational Ability Coordinates & Constraints
Calibration Parameters Expected Value Constraint Negativity Constraint Xi = Yi