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ECON4450
11/15/2010
Introduction
As with many social science topics, examples provide a clear understanding of the topic
itself. Game theory is no different. The following paper provides an explanation of basic game
theory topics through the use of a soccer game as an example. Through this single scenario, basic
topics such as strategies, types of models, rationality, and equilibriums can be demonstrated
clearly. Most importantly, the scenario illustrates how game theory provides tools to explain and
model social behavior.
Scenario
In 1994, the Shell Caribbean Cup (SCC), an international soccer tournament for Caribbean
nations, was taking place in Trinidad and Tobago. The first part of the tournament was called the
group stages. There were six groups consisting of three teams each. Each team in a group played
one game against each opposing team in the same group. The team with the best record in each
group would advance to the next stage of the tournament.
Group one of the tournament included Barbados, Grenada, and Puerto Rico. Prior to the final
group one match played between Barbados and Grenada, Barbados had beaten Puerto Rico 1-0
while Grenada had similarly won against Puerto Rico, 2-0. Therefore, Puerto Rico was
mathematically disqualified from advancing while Grenada and Barbados both had one win prior
to playing each other in the deciding match over who would advance. However, since Grenada
had scored more goals than Barbados coming into the final match, Barbados would need to win
the game against Grenada by at least two goals or they would not advance and Grenada would
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(even though they may have lost that game). On the other hand, Grenada simply needed to win
the game to advance. In the event that a tie occurred, the tournament officials had determined
that the teams would play for two additional fifteen minute halves. The format of this extra time
was “golden goal.” This meant that the team who scored first in extra time would win the game.
This format was similar to other International Federation of Association Football (FIFA)
established tournaments; however, there was one major difference here: the “golden goal” scored
in extra time would count as two goals. This enabled Barbados to advance if it were to score the
golden goal in extra time.
After nearly 80 minutes of play, Barbados looked likely to progress to the next stage of the
tournament because they were winning 2-0 and had the necessary two goal lead that they needed
to advance. Nonetheless, with seven minutes of play left in the game, Grenada scored a goal to
make the game 2-1 and put the Barbadians in an awkward situation. Scoring another goal with
seven minutes of the game left was a daunting task that seemed very unlikely. However, because
of the rules set up by the tournament officials, scoring a goal was not Barbados’ only option that
would give them a chance of winning the game and progressing to the next stage of the
tournament. They also could score a goal on themselves to tie the game and force it to extra
time. Game theory provides adequate tools to model how this scenario may ensue; however,
prior to being able to use these tools it is necessary to understand basic definitions.
Players
A player in a game is anyone “who [interacts] in that environment” of strategic
interdependence (Harrington 8). Strategic interdependence is “what is best for someone depends
on what someone else does” and “a situation in which what one person does affects the wellbeing of others” (Harrington 2). Therefore, a person who may be affected by the outcome of a
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strategic situation but does not act in that situation is not a player. Consequently, the only players
in this scenario are Grenada and Barbados.
Players are defined in two ways: by their preferences and beliefs. Beliefs are formed through
prior experiences, called experiential learning, and by thinking about how the other players are
thinking, called simulated introspection. Preferences are complete (players either prefer one
option over another or are indifferent between the two, transitive (if one prefers a to b and b to c,
then he prefers a to c) , and exhibit non-satiation (more is better). Preferences are not criticized
or questioned; they are simply taken as is.
Objectives
Preferences and beliefs are important characteristics of players because they dictate their
objectives, which ultimately define the payoffs. In the SCC, Grenada’s objective is to advance to
the next stage of the tournament without going into overtime. Barbados’ objective also is to
advance to the next stage of the tournament (even if it means forcing the game into extra time).
Extensive Form
In modeling a social situation using game theory, we use simple models that capture the
details of the situation. The extensive form is one such model. The extensive form details the
actions available to each player at specific locations in a strategic situation. Figure 1 shows the
SCC scenario modeled in extensive form. The order of the game is from top to bottom, left to
right. This type of model often is called a decision tree. An event in a scenario where a player
has to take an action or make a decision is represented by a dot on the tree and is called a
decision node. Each branch extending from a node represents a particular choice or action. For
example, the second dot in the second row of dots in Figure 1 is Grenada’s second decision node
and they have two branches representing the two choices they must decide between, scoring an
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own goal or scoring a regular goal. The first node and last node(s) of a decision tree are labeled
initial and terminal nodes, respectively.
An important feature of the extensive model is that it sequentially models the game and
depicts whether or not a player knows the decision an opposing player has taken before or
whether the decisions of the two players are made simultaneously. In some situations players
know all information about others’ choices. At others times, there is a lack of information for a
player. If this is the case then imperfect information exists in the situation. Imperfect information
is a situation where one player does not know what other players have done at a point in the
game. For example, if Barbados scores an own goal, Barbados then must make a decision on
which goal to defend while Grenada simultaneously chooses which goal to score on. In this
example, when Barbados makes their second decision, they do not know what Grenada has
chosen to do. In order to model this situation, information sets are used. Information sets contain
at least one decision node and can be defined as “all of the decision nodes that a player is
incapable of distinguishing among” (Harrington 27). Information sets often are depicted as a
dashed line connecting two decision nodes that a player cannot distinguish between. Because
Grenada and Barbados must make a decision simultaneously if Barbados chooses to score an
own goal at the initial node, this situation is represented by an information set containing the two
decision nodes resulting from Grenada’s choices of scoring a regular or own goal. The
information set illustrates Barbados’s lack of knowledge as to what Grenada has chosen at their
previous node. Consequently, Barbados is unable to tell what decision node they are at.
It is crucial that in modeling any scenario important assumptions are made. First, strategic
situations are common knowledge. This means everyone interacting in the game knows the game
is being played, and that each person knows the other people know and so forth. Secondly, we
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assume perfect recall; a player who has gained information at some point will perfectly
remember it later. Therefore, if Barbados reaches the point where it must decide between
defending its own goal or Grenada’s, because of perfect recall, they know that at the initial node
they scored a goal on themselves and Grenada either is trying to score a regular or own goal.
With the key concepts of the extensive form model explained, modeling the SCC scenario is
relatively a simple task. The game starts after Grenada has scored their first goal. Since the ball
is now in possession of Barbados, they have two feasible actions: try to score a goal or score an
own goal. Try is left out of scoring an own goal because it is assumed Barbados will score an
own goal if it tries because the task is relatively simple at this point. If Barbados tries to score a
goal, then Grenada has to make a decision, which is represented by Grenada’s first node in
Figure 1. At this node, Grenada has two options: hold back and defend or try to score a goal. On
the other hand if Barbados chooses to score a goal, the game takes a different route. Grenada is
now faced with two different options: either trying to score an own goal or trying to score a
regular goal. Here it is assumed that Grenada is trying in both cases because even if they try to
score an own goal there is a possibility Barbados may be guarding the other team’s goal! Next it
is assumed that Grenada’s decision to try to score an own or regular goal is taken simultaneously
as Barbados’ option of choosing which goal to defend (because at this point they want the game
to tie and go into overtime). Because this is a simultaneous decision, imperfect information exists
so Barbados’ second and third nodes are depicted as one information set since they do not know
which node they are at if they originally chose to score an own goal.
Payoffs
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At the bottom of each initial node the playoffs for each player are offered, usually in a
quantifiable method. The payoffs need not represent a specific object such as money; they only
serve as a rank with meaningful distances for preferences for different outcomes. The payoffs are
listed by from top to bottom in order of players. The payoffs result from the players’ objectives.
In the SCC scenario, the payoffs for Grenada and Barbados in the six different scenarios follow
directly from the objective provided above.
Strategy
Modeling the SCC scenario in extensive form provides many benefits to understanding the social
behavior; however, it is important also to derive the strategic form model for the game because
the strategic form has unique benefits as well. Therefore, it is crucial to understand strategies as
their use is necessary in the strategic form. A strategy can be defined as “a fully specified
decision rule for how to play a game” (Harrington 34). Strategies not only cover some parts of
the game, strategies provide an action to each contingency. A strategy can be illustrated using the
extensive form as a rule that prescribes a specific action at each information set. The strategy set
is the set of all strategies and a strategy profile is a list of one strategy from each player in the
game. A strategy is written based on the decision nodes, from top to bottom, left to right. For
example, a strategy for Barbados might be, score own goal/defend own goal. Because they only
have two information sets, Barbados’ strategy only prescribes two actions. Similarly, Grenada
has two information sets so a strategy for them is attack to score/score own goal. Notice how the
strategy details an action at each node, even if that node is not realized during play.
Strategic or Normal Form
The reason strategies are important is that they are used in modeling the scenario in the
second most important form, strategic form. Unlike the extensive form, the strategic form is
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typically a matrix with rows and columns for two player games. The first player is the row player
and the second player is the column player. The strategy set of each player is listed in the rows or
columns, one strategy in each. See Figure 2. The intersection of a row and column represents a
strategy profile or a possible outcome of a game. For games with three players, the third player
would be considered the box player and each two player matrix would be replicated the same
number of times for each possible strategy available to the third player. Therefore, an
intersection of a column and row in a specific box would represent a strategy profile or outcome.
Games with n-tuples players are diagramed in a similar fashion.
To diagram the SCC scenario in strategic form we need the strategy set for each player. Next,
we set up the matrix as in Figure 2. The order of strategies does not matter. Finally, we add the
payoffs in each box for the outcome the specific strategy profile represents. Using these three
steps, the SCC scenario is modeled in strategic form. Although a scenario can be modeled
directly to strategic form, it often is easier to model it first in extensive form and then in
strategic form. Notice how multiple strategic profiles result in the same outcome. This is because
strategies contain rules for every possible contingency even if some contingencies are not
realized. Therefore, strategies that have the same decision at one node but different at another, as
long as the strategy of the other player results in only that first node being played, then the
outcome will be the same for both strategies because the second node is not reached.
Nash Equilibrium
While modeling a scenario is important, the most important ability of social sciences is the
ability to explain or predict behavior. That is where equilibriums are important. In particular, the
most important equilibrium is Nash Equilibrium (NE). Before defining NE, rationality must be
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defined. In game theory, all players are assumed to be rational by maximizing their payoffs
“given [their] beliefs about what other players will do” (Harrington 90). Another assumption is
that rationality is common knowledge. In order to define NE another important assumption is
made; “each player’s beliefs about the strategies used by the other players are true” (Harrington
90). With these assumptions established, NE can now be defined. A Nash equilibrium is a type
of a strategy profile that satisfies the following: each strategy in the strategy profile maximizes a
player’s payoff, given the other players’ strategies in that profile. A NE is a solution to a game
because it is a rational outcome for all players. To solve for a NE, all players’ best replies for
each strategy used by the other players must be found. A strategy profile that contains strategies
that are all best replies is a NE. A best reply is the strategy that maximizes a player’s payoff
given the strategy of the other player. So, if the second player in a two-player game has two
strategies, the first player has at least two best replies. It is possible for a player to have two best
replies for one given strategy of the player. In this case, the player is indifferent between those
two strategies. Using this algorithm, solving for NE in the strategic form is not difficult. For
example, to find the best reply for Grenada to Barbados’ strategy of score own goal/defend
Grenada’s goal, hold Barbados strategy fixed and the best reply for Grenada is the strategy that
maximizes Grenada’s payoff in that given row. The rest of the best replies are found similarly. A
NE is a strategy profile in which both strategies are best replies, so a box in the normal form
matrix that contains two best replies is a NE. Nonetheless, there is no pure strategy NE in this
scenario. To illustrate what a NE equilibrium is, if the payoffs for the strategy profile
(defend/own goal, score own goal/defend Grenada’s goal) had payoffs of 3,4 then this would be
a NE equilibrium because both strategies would then be best replies. Although the SCC scenario
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does not have a pure strategy NE equilibrium, game theory provides another useful tool to find a
solution to this scenario.
Iterative Deletion of Strictly Dominated Strategies
Iterative Deletion of Strictly Dominated Strategies (IDSDS) is a procedure that can solve
games and/or rule out strategies. A strategy that strictly dominates another strategy provides a
greater payoff regardless of the strategies chosen by the other players. A strategy that is strictly
dominated by another strategy is called strictly dominated. A strategy weakly dominates another
strategy if it receives a greater payoff for some of the strategies chosen by the other players and
at least as good as other strategies given what the other player has chosen. With these definitions,
it is obvious a rational player will never use a strictly dominated strategy because since the
player isrational he will maximize his payoff. Therefore rationality explains that a player will
never choose a strictly dominated strategy because he can always do better with another strategy.
IDSDS uses this important concept to help solve the game. IDSDS is a procedure that relies on
the assumption that rationality is common knowledge among players. The procedure is done in a
series of steps. The first step is to delete all strictly dominated strategies present in the game.
Next, strategies that are now strictly dominated are deleted. This process is repeated until no
more strictly dominated strategies exist. In some instances IDSDS results in the deletion of all
strategies except for one for each player. In this case, only one strategy profile is left and it is the
outcome of the game. If this is the case the game is said to be dominance solvable. In other
instances, IDSDS either only allows for the deletion of a few strategies or cannot delete any
strategies at all (because there are no strictly dominated strategies). It is important to note that
weakly dominated strategies are not deleted because they have at least one outcome that is at
least as good as another strategy. Also important is the fact that all NEs survive IDSDS because
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each strategy in a strategy profile that is NE is a best reply, meaning it is at least as good as
another strategy. Therefore NE cannot contain strategies that are strictly dominated. Thus NE
comply with the rules set by IDSDS and are logical solutions to the game. In fact, if a game is
dominance solvable, that single solution is a NE because both strategies in that profile are best
replies.
Because it is easy to compare strategies in strategic form, IDSDS frequently is performed in
this form. The SCC scenario is not dominance solvable. IDSDS cannot be relied upon to provide
a single solution to the game; however, it can be used to help in solving the game because two
strictly dominated strategies are present. Using the definition of a strictly dominated strategy, it
is obvious Barbados’ two strategies try to score goal/defend own goal and try to score
goal/defend Grenada’s goal are both strictly dominated by Barbados’ other two strategies
because score own goal/defend own goal and score own goal/defend Grenada’s goal always
provides strictly greater payoffs. Therefore, we can delete those two strategies. Because Grenada
does not have any strictly dominated strategies the game is now a 2x4 matrix as in Figure 3.
With this new game, no new strictly dominated strategies are present so IDSDS cannot be used
any further. The next step is to use the new game created by IDSDS to find a unique NE solution
for the game.
Mixed Strategy Nash Equilibriums
Since there is no pure strategy NE, the model does not explain which strategy will be taken.
Nonetheless, there is still a unique solution to the SCC scenario, and it helps that IDSDS rules
out Barbados trying to score a goal because now the 2x4 matrix is more manageable. Because no
pure strategy NE exists, it is a reasonable assumption that players will vary their decisions
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according to certain probabilities in the hopes of outguessing the other player. Intuitively, it
makes sense that players would randomly pick different decisions according to a probability rule.
For example, Barbados would not always want to choose the strategy score own goal/defend
own goal “because such predictability would induce” Grenada to always choose the strategies
defend/own goal or attack/own goal which would give Grenada its highest payoff given
Barbados strategy. However, Barbados would receive the lowest payoff, a punishment for
predictability.
Prior to continuing, it is necessary to clarify basic probability concepts. First a random
outcome is an unpredictable outcome and a random variable is a random outcome that takes
certain values each. Each value of the random variable is associated with a probability. A
random variable value’s probability is the number of times that value would occur if the random
outcome were to occur an infinite number of times. A probability distribution for a random
variable is all the possible values of the variable and their associated probabilities. Two
outcomes are said to be independent if neither outcome has an effect on the probability of the
other. Probabilities are always between 0 and 1 (inclusive) and the sum of the probabilities for all
values of a random variable must equal 1. If a random variable only takes on two different values
and one of the values has a probability of p, it must be the case the other probability is 1-p
because their sum must be equal to one. A random variable’s expected value is defined as “the
weighted sum of the possible realizations of that random variable, where the weight attached to a
realization is the probability of that event” (Harrington 184).
It is possible to derive a unique NE to the SCC scenario. Because there is no pure strategy NE
and players vary their actions, the game is now randomized because players choose strategies
according to a probability. A strategy is no longer one specified decision rule for each node; it
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contains all strategies and different probabilities with which those strategies are played This is
called a mixed strategy because the players are “mixing” up strategies according to a probability
distribution. Because a strategy is randomized, a strategy profile is also randomized. In order to
evaluate the payoffs, the expected value of the payoffs from a strategy profile is derived and used
as the payoff to compare this option.
With mixed strategies is it now possible to find a NE, in this case a mixed strategy NE. A
mixed strategy NE is a mixed strategy profile where each player has maximized his or her payoff
given the mixed strategy used by the other players. Because the possible probabilities for any
value of a random variable are infinite, it may appear daunting to find the specific probabilities
for mixed strategies that satisfy the criteria for a mixed strategy. However, a few facts reveal a
simple method for deriving those probabilities. First, each players expected payoff from a
strategy is solely dependent on the mixed strategy of the other players. (The proof is left out).
Therefore expected values for a player’s strategies can be evaluated according to the
probabilities that the other players play certain outcomes. See the derivation of the expected
values in Figures 3 for an example. With the expected value from all strategies of a certain
player, those values are set equal to each other in order to solve for the particular value of the
probability that makes the player indifferent between all of those strategies. This ensures that any
mixed strategy the player plays is always a best reply, because he will always be indifferent since
that payoff is solely dependent on the other player’s mixed strategy. Doing this for all players
provides the mixed strategies for all players and a mixed strategy can be represented as the
probability distribution of a each player’s mixed strategy, but if there are only two strategies for
each player, by convention, only the players’ first strategy’s probability is provided because the
other probabilities can be found easily. In the SCC scenario, Barbados has two strategies so they
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have two probabilities that represent their mixed strategy. Similarly, Grenada has four strategies
so they have four possible probabilities that represent their mixed strategy. In this case, the
mixed strategy NE derived in Figure 3 is ((1/3)/ (2/3), .25,.25,.25,.25). This means that a third of
the time Barbados will score an own goal and defend their goal and the rest of the time they will
score an own goal and defend Grenada’s goal. It also means Grenada will employ each of its four
strategies an equal amount of time. In this mixed strategy NE, Barbados’ expected payoff is 8/3
while Grenada’s is 2.5. Because this is a mixed strategy NE, this is the greatest payoff for each
player given what the other player’s mixed strategy is. Therefore, if we have modeled the
scenario correctly we would expect each player to choose each decision according to the
probability distribution specified in the mixed strategy NE.
Conclusion
Thus game theory has provided a unique solution to this peculiar final group stage game of the
SCC: the mixed strategy NE calculated above. Nonetheless, the model’s accuracy cannot be
tested without knowing what actually happened in the game. Well, it turns out Barbados
purposely scored an own goal and then Grenada proceeded to mix between trying to score on its
own goal and Barbados’ goal while Barbados simultaneously switched between defending their
goal and Grenada’s goal. After 90 minutes the teams were still tied, so they went to overtime. In
overtime, Barbados’ scored the only goal and advanced to the final at the expense of Grenada.
The Grenada coach had this to say after the game, "The game should never be played with so
many players on the field confused. Our players did not even know which direction to attack; our
goal or their goal. I have never seen this happen before. In football, you are supposed to score
against your opponents in order to win, not for them.”
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References
Harrington, Joseph. Games, Strategies, and Decision Making . 1st ed. New
York: Worth Publishers, 2009. Print.
"Barbados vs. Grenada in '94: The Most Bizarre Match Ever." Bleacher
Report. N.p., 29 Oct 2008. Web. 16 Nov 2010.
<http://bleacherreport.com/articles/74831 -barbados-vs-grenada-in94-the-most-bizarre-match-ever>.
T a l w a l k a r , P r e s h . " T h e w e i r d e s t s o c c e r m a t c h a n d g a m e t h e o r y. " M i n d
Your Decisions. 16 Jun 2010. Web. 16 Nov 2010.
< h t t p : / / m i n d yo u r d e c i s i o n s . c o m / b l o g / 2 0 1 0 / 0 6 / 1 6 / t h e - w e i r d e s t s o c c e r - m a t c h - a n d - g a m e - t h e o r y/ > .