Lecture11.ppt

School of Computing &
Information
INFSCI 0530: Decision
Making in Sports
Fall 2021

Information
Game Theory
• Game theory deals with mathematical models that
describe the strategic interaction between
decision makers
– These models assume that the decision makers are
rational
– This is not necessarily true in the real-world but it
gives us a good starting point when making our own
decisions

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Types of games
• There are several types of games depending on the
strategic setting
– Cooperative – vs- non-cooperative
– Symmetric – vs- asymmetric
– Zero-sum – vs – non-zero-sum
– Simultaneous – vs – sequential
– …

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Zero-sum game
• A two-player zero-sum game (TPZSG) involves:
– Two players
– The gain of the winning player is equal to the loss of
the losing player
• Every player has a set of strategies S that they can
choose from
• Depending on the strategies s1 and s2 chosen by
each player there is a specific payoff for each
player p(s1,s2)

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Zero-sum games
• These payoffs are summarized through a payoff
matrix P
– Rows represent the strategies of player 1 and columns
represent the strategies of player 2
– The element Pij represents the payoff for player 1
when they choose strategy i and player 2 chooses
strategy j
• Since this is a zero-sum game the payoff for player 2 is
-Pij
• The payoff matrix can be estimated through data

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Zero-sum games
• A player can choose between a pure strategy or a
mixed strategy
• Nash-equilibrium of a game is a choice of
strategies (mixed or fixed) that no player has any
incentive to switch unilaterally to another strategy
– If they do switch, while the other player sticks to the
Nash equilibrium strategy they are going to obtain
lower payoff
– Nash equilibrium does not necessarily provide the
global maximum payoff to players

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Zero-sum games
• An important theorem from John von Neumann
states that a TPZSG always has a – possibly mixed
strategy - Nash equilibrium
– Some requirements for the theorem:
• Game is of complete information, i.e., we know all the
moves available to us and the opponent, as well as, the
payoffs
• Game is finite, i.e., the available strategies are finite and
the game ends after a finite number of moves

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Finding the Nash equilibrium
• Let’s start by considering a simple TPZSG with the
following payoff matrix
• The Nash equilibrium represents a strategy (for
the row player) that essentially maximizes the
minimum possible gain by choosing this strategy
regardless of the choice of the other player
• If we look only at pure strategies the Nash
equilibrium matches the saddle point of the payoff
matrix

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Saddle point
• A saddle point for a matrix is any element that is
both the minimum of its row and the maximum of
its column
• Not all matrices have a saddle point, but if the
payoff matrix has one, this is also a Nash
equilibrium
– Why?

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Mixed Nash equilibrium
• If the payoff matrix does not have a saddle point,
then the TPZSG has a mixed Nash equilibrium
• The mixed Nash equilibrium is essentially a
probability distribution over the available
strategies
– It tells us how often we should employ each strategy
(at random)
• In order to identify this mixed Nash equilibrium
we will follow a similar line of thought as with the
saddle point

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Run – vs - Pass
• Let’s see a specific example to understand the
notion of TPZSG better and also see how we can
calculate the mixed Nash equilibrium.
• The two players are the offense and the opposing
defense in an NFL game

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Run – vs – Pass
• Let’s assume that the offense chooses to run with a
probability q  it chooses to pass with a prob 1-q
• What is the expected gain we have from this mixed
strategy assuming the opponent plays a fixed
strategy (i.e., always run defense or always pass
defense)
– Against run D: q*(-5) + (1-q)*10 = 10 – 15*q
– Against pass D: q*5 + (1-q)*0 = q*5

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Run – vs – Pass
• Which fixed strategy is the defense going to
choose?
– The one that yields the minimum of the two expected
gains {10-15q, 5q}
• Consequently, the offense should choose q in such
a way that the minimum above is maximized

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Run – vs – Pass

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Run – vs – Pass
• So, for the offense if they choose to run 50% of the
time, then whatever strategy the defense chooses
they are guaranteed an expected yardage of 2.5
• Even though passing is a much better option over
all the optimal strategy – with these fictitious
payoffs – is to run 50% of the time
– This provides a good rationale why teams mix up their
running and passing calls

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Run – vs - Pass
• What about the defensive side ?
• Let’s assume they play run D with probability x
and they play pass D with probability 1-x
• The expected gains for the offense (that the
defense wants to minimize) are:
– Against running play: x*(-5)+(1-x)*5 = 5-10*x
– Against passing play: x*10+(1-x)*0 = 10*x
• How does defense choose x?

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Run – vs - Pass

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Run – vs - Pass
• The minmax is obtained with x = ¼
– The defense defends the pass 75% of the time and the
run 25% of the time
– This is because the pass play is better than the run on
average
• This also creates a feedback and leads the offense to
calling run 50% of the time in this game, even though
the passing plays are more efficient

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Football paradox
• In the homework you will work on a similar
problem:
• We get a new RB

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The Corner-3 Game
• We saw in previous lectures that the main reason
behind the efficiency of corner-3s in the NBA is
the fact that they are assisted at a higher rate and
not that they are closer to the basket
• One of the reasons is the imbalance created in the
defense by drive-n-kick action that generates more
than half of these shots
– Corner 3 shooter anchored at his spot waiting for kick
out pass
– Corner 3 defender has to decide whether to help the
penetration or stay with his man

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The Corner-3 Game
Zero-sum game
Defense
Offense
Payoff matrix

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The Corner-3 Game
• How do we fill out the payoff matrix?
– This matrix can be customized to each situation
(team, players, situation etc.)
– Data can help with quantifying the payoff of each
situation.
• We will consider the league-average case to just
get an idea of what is going on and what is the
Nash equilibrium in these situations

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The Corner-3 Game
When offense chooses
kick-out pass
When offense chooses drive
Impact of primary defender Impact of
double team

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The Corner-3 Game

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The Corner-3 Game
While the expected position of the defender in actual games is
similar to what one would expect from the Nash equilibrium,
teams reach this through a different mix strategy!

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Penalty kicks and game theory
• Penalty kicks offer another area where game
theory can be applied
• Again we have a TPZSG, where the kicker and the
goalie have the same set of strategies
– Kick/Jump left, right or middle

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• Depending on the values of the payoff matrix the
Nash equilibrium will be different
• The solution to this game shows that if:
– Both the kicker and the goalie never choose middle
– Thus, kickers never kick in the middle unless if the
probability of scoring (πL, πR) is large enough

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• The heterogeneity of the matches (i.e., the payoff
matrix is not the same for every team, players,
league etc.), this will create a selection bias in that
the aggregate scoring probability should be
large for kicks to the center
– I.e., kickers will choose center when this scoring
probability is high
• This pattern is indeed observed in the data
– Also what is observed and expected from the
heterogeneity is that the kicks to the center are more
than the kicks for which the goalie stayed at the center

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Non-constant utility
• In the examples until now the payoff matrix is
constant
• However, you can easily envision situations where
the utility from a strategy depends on the
frequency with which it is used
– For example, player skill curves in the NBA

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• Another example is the efficiency of passing game
in the NFL

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• For example, the passing efficiency of a team is a
declining function of its utilization
– Higher utilization, lower efficiency
– Efficiency is measured based on the notion of
expected points
• While running might still be less efficient
compared to passing 100% of the time, passing
100% of the time will not yield the maximum
possible expected yardage.

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• This correlation remains even when we control for
other variables

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• While the question of how much should we run is
not a simple one and we can only approximate its
answer, it should be clear that passing the ball all
the time will have diminishing returns
• For passing utilization  1 an individual passing
play might still provide higher efficiency as
compared to an individual rushing play
– However, the diminishing returns mean that we can
achieve a higher overall efficiency per play

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• With u being the passing utilization, r being the
rushing rating of the offense and p being the
passing efficiency of the offense based on the
regression model we need to maximize:
• For an average rushing team (r approximately 0.1)
we get:
– u = 0.3 for a bad passing offense
– u = 0.47 for an average offense
– u = 0.63 for a great offense

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• What is an assumption that we have implicitly
made in the above analysis?
• That the rushing efficiency does not change with
utilization which is large
– This is largely true based on data

Lecture11.ppt

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