The authors present a simple and effective method for improving student comprehension of mixed strategies. After reviewing available texts, and with some trial and error, we feel that this method eases the teaching of the information and increases the students' ability to understand and retain concepts. Through a simple alteration to the normal form of a game, students can clearly see the results of employing mixed strategies, practise calculating expected payoffs, visualise the accompanying 'indifference' in the payoffs, and recognise the mixed strategy Nash equilibrium in a familiar way.

Evan Moore

Kenneth Garrett

English

Research Papers in Economics

79
Teaching Mixed Strategy
Nash Equilibrium to
Undergraduates
Kenneth Garrett and Evan Moore 
Abstract
The authors present a simple and effective method for improving student
comprehension of mixed strategies.After reviewing available texts,and with some
trial and error,we feel that this method eases the teaching of the information and
increases the students’ability to understand and retain concepts.Through a simple
alteration to the normal form of a game,students can clearly see the results of
employing mixed strategies,practise calculating expected payoffs,visualise the
accompanying ‘indifference’in the payoffs,and recognise the mixed strategy Nash
equilibrium in a familiar way.
Introduction
Understanding the concept of an optimal mixed strategy is not an easy task for the
inexperienced undergraduate student.Anecdotal evidence suggests that the
approaches used in a variety of textbooks to present this material leave students
feeling that ﬁnding optimal mixed strategies and the resulting Nash equilibria is
nothing more than a mathematical exercise.We propose a simple method that
reveals how a mixed strategy Nash equilibrium (MSNE) is similar to Nash
equilibrium in pure strategies.Additionally,the proposed method aids students in
grasping the concept of ‘indifference’for each player when faced with an optimal
mixed strategy.Our approach is easy to implement with a simple modiﬁcation to
the normal form of a game,and we have found it to be quite effective in helping
students to master the concept of mixed strategies.
Mixed strategies involve a player randomly choosing among pure strategies
according to given probabilities.Bernheim and Whinston (2008) deﬁne a mixed
strategy as a player using,‘a rule to randomize over the choice of a strategy’.TheInternational Review of Economics Education
80
deﬁnition given by Pindyck and Rubinfeld (2005) may be clearer,as they deﬁne a
mixed strategy as a ‘strategy in which a player makes a random choice among two
or more possible actions,based on a set of chosen probabilities’.The value of mixed
strategies should not be underestimated.There are a number of motivations given
in undergraduate textbooks regarding the need for,and use of,mixed strategies.
Pindyck and Rubinfeld (2005) state that,‘there are games… in which a pure
strategy is not the best way to play’.McCain (2004) describes the need for a mixed
strategy when the player would beneﬁt from being ‘unpredictable,so that the
opposition cannot guess which strategy is coming and prepare accordingly’.
Bernheim and Whinston (2008) support this by motivating their discussion on
mixed strategies with,‘the key to success… is unpredictability.…The most obvious
choice is to make choices randomly’.Baldani et al.(2005) write:‘Not all games have
pure strategy Nash equilibria in which each player chooses a single strategy with
probability one.There are important economic applications that only have
solutions in mixed strategies in which a player randomizes by choosing the
probabilities for playing the possible pure strategies.’
There is a wide variety of instances of mixed strategies in the ‘real world’.These
include uses in games such as ‘Rock-Paper-Scissors’to the world of sports.Examples
in sports include baseball pitchers alternating between various pitches,such as a
curveball,sinker and fastball,to keep hitters off-balance,or football offences varying
between the passing and running attacks to keep defences ‘honest’.Two articles in
the American Economic Review look at mixed strategies in sports;Chiappori et al.
(2002) investigate the use of mixed strategies during penalty kicks in professional
soccer matches and Walker and Wooders (2001) review mixed strategies employed
by servers to opponents’forehands or backhands in professional tennis matches.A
nice resource for articles and movies that mention or involve mixed strategies,and
game theory in general,is www.gametheory.net.It includes articles involving the
concept of mixed strategies in dealing with terrorism,tax cheats and the Internal
Revenue Service,playing poker,quantum mechanics,and how beating the S&P is
like a game of ‘matching pennies’.1These articles are from a range of sources
including magazines such as the Economist,Nature,Newsweek and Slate,to the news
divisions of ABC and CBS,and a variety of newspapers.There are also instances of
mixed strategies in ﬁlms.John Kay (1994) writes:‘But the greatest paradox of all is
that it is often best to adapt random behaviour in chicken games.It is possible to
write down the mathematics of the problem faced by the two groups of boys in
Stand By Me,and to show that in general a good solution for each is to swerve
sometimes and to stick sometimes.You can even calculate how often you should
swerve and how often you should stick.’Other examples of ‘Chicken’can be found in
the ﬁlms Rebel without a Cause and Footloose.A game of the ‘Battle of Wits’is in the
The Princess Bride,a discussion of which is presented in Bernheim and WhinstonTeaching Mixed Strategy Nash Equilibrium to Undergraduates
81
(2008).Given the wide range of applications where the uses of mixed strategies
apply in decision making,we feel an improved understanding of the topic is useful
to undergraduate economics students.
Textbook approaches
We selected a variety of universities to see which textbooks were used in
Intermediate Microeconomics,Managerial Economics and Mathematical
Economics.We used courses from Massachusetts Institute of Technology,University
of California at Berkeley,Auburn University,University of Georgia,Wake Forest
University,University of New Mexico,Amherst College and University of Nebraska at
Omaha.2We believe that many other colleges and universities are likely to use
these,or similar,textbooks in their courses.The goals of our review were to
investigate:1) the mathematical method for determining the optimal mixed
strategies;and 2) how the end results are visually presented in the textbook.
The textbooks we reviewed fell into one of two categories regarding the
methodology used to determine the optimal mixed strategies:
1 Determining the mixed strategies algebraically,generally by solving for the
probability of one strategy,p,and the probability of the other strategy choice as
a function of the ﬁrst strategy,i.e.1-p,and thus making the opponent indifferent
(Hay et al.;Nicholson and Snyder;McGuigan et al.;Neilson and Winter;Pindyck
and Rubinfeld;Varian).
2 Determining the mixed strategies by taking the derivative of a linear function
that represents players’strategy expectations.This method,while not difficult,
was the least user-friendly and it is easy to see that it gives the same results as
the simpler algebraic method (Baldani et al.,2005).
We did not ﬁnd any textbook approach that includes our proposed method for
improving student learning and comprehension of the concept of mixed strategies
and the resulting Nash equilibrium.
In the classroom
To get the students engaged in the concept of mixed strategies,we have them
group into pairs and play a few rounds of a standard ‘matching pennies’game.We
provide them with very little explanation of mixed strategies other than broadly
discussing making the other player indifferent between strategy choices and we do
not show any technique to solve for an optimal mixed strategy at this point.
Nevertheless,it typically does not take the students long to arrive at the optimal
mixed strategy,i.e.probability weights of 
1⁄2 for each of the strategies.In someInternational Review of Economics Education
82
instances we followed this with the pairs playing a few rounds of ‘Rock-Paper-
Scissors’and the students quickly conclude that the optimal mixed strategy
involves attaching a probability weight of one-third to each strategy.After playing
these zero sum games,we introduce a simple 2x2 non-zero sum game,such as the
one shown in Figure 1.For the purposes of this paper we show each player’s best
response to the other player’s choice by underscoring the payoff.Note that this
game does not have a Nash equilibrium in pure strategies,as evidenced by no
outcome having both payoffs in a cell underscored.
We then ask the students to propose a few possible probability weights for
equilibrium mixed strategies,reminding them that the aim of a player’s mixed
strategy is to make the other player indifferent between his strategy choices.
Frequently they respond with weights of 
1⁄2 for each strategy.We then add an
additional row for player 1 and column for player 2 to the original game,labelling
each strategy to reﬂect the selected weights.This is shown in Figure 2.The expected
payoffs in the new cells by row are calculated using the weights of player 2’s mixed
strategy.For example,consider the expected payoffs in the cell resulting from
player 1 playing A and player 2 playing the mixed strategy of 
1⁄2 C and 
1⁄2 D.Player
1’s expected payoff from playing A is 
1⁄2*4 + 
1⁄2*12 = 8.Player 2’s expected payoff
when player 1 plays A is 
1⁄2*25 + 
1⁄2*5 = 15.These expected payoffs are listed in the
cell in the upper right corner of Figure 2.Similarly,the expected payoffs from player
2 playing a strategy versus player 1’s mixed strategy are calculated using weighted
values of the payoffs by column.3 At this point we once again identify each player’s
best response for this 3x3 matrix using underscores.We then point out to the
students that there is no Nash equilibrium with the 
1⁄2 probability weights.
Once again we reintroduce the notion of indifference,and that a player is
indifferent between strategy choices if the expected payoff is the same for that
player.We note that this means player 1 will have the same expected payoff from
playing A,B or the optimal mixed strategy when facing the optimal mixed strategy
Figure 1:A 2x2 non-zero sum game
25
4
5
12
10
16
15
8
CD
A
B
Player 1
Player 2Teaching Mixed Strategy Nash Equilibrium to Undergraduates
83
choice for player 2,i.e.all of player 1’s expected payoffs will be the same in the
optimal mixed strategy column for player 2.Likewise,we take the time to point out
that player 2’s expected payoffs will be the same when facing the optimal mixed
strategy from player 1,i.e.all of the payoffs to player 2 are identical in the same
mixed strategy row for player 1.
At this point in class we introduce how to solve for the optimal mixed strategy
choices using the common textbook techniques.This is done by equating a player’s
expected values from each of her strategy choices,i.e.player 2 is mixing between
strategies C and D to make player 1 indifferent between playing strategies A and B,
while player 1 is mixing between strategies A and B to make player 2 indifferent
between strategies C and D.We provide the solution for the game in Figure 1 using
the standard algebraic approach.We use this approach because generally the
students seem more comfortable using algebra and because six of the seven
textbooks reviewed that cover mixed strategies use an algebraic approach.4The
solution using calculus is provided in the appendix.
Player 2 mixing to make Player 1 indifferent;Player 1 mixing to make Player 2
indifferent
EV(A) = EV(B)  EV(C) = EV(D)
4P2C+12(1-P2C) = 16P2C+8(1- P2C) 25 P1A+10(1- P1A ) = 5 P1A +15(1- P1A )
4P2C+12-12P2C = 16P2C+8-8P2C 25 P1A +10-10P1A = 5 P1A +15 -15 P1A
12-8P2C = 8P2C+8  15 P1A+10 = 15-10 P1A
4 = 16P2C 25 P1A = 5
P2C =
4⁄16 =
1⁄4 P1A =
1⁄5
P2D = 1 – P2C =
3⁄4 P1B = 1 – P1A = 
4⁄5
Figure 2:Including mixed strategies with probability weights of 
1⁄2
25
4
5
12
15
8
10
16
15
8
12 1/2
12
17 1/2
10
10
10
13 3/4
10
CD
1⁄2 C,
1⁄2 D
A
B
1⁄2 A,
1⁄2 B
Player 1
Player 2International Review of Economics Education
84
We follow this by adding the newly found optimal mixed strategies to Figure 1 to
create Figure 3.We then calculate the new expected payoffs for each of the cells
involving the mixed strategies.We stress to the students that,when properly
weighted,the 3x3 game now clearly shows each player’s indifference to the other
players’mixed strategy choice.
Once again we underscore the best responses for each player.This results in the
MSNE as shown in the bottom right cell of Figure 3.We ﬁnd that this simple
addition,including the mixed strategies in the normal form,enhances the ability of
students to grasp the concept of indifference,calculate expected payoffs,and
visualise the MSNE in a similar way to Nash equilibrium in pure strategies.5
Conclusion
We present a simple and effective method for improving student comprehension
of mixed strategies.After reviewing available texts,and with some trial and error,we
feel that this method eases the teaching of the information and increases the
students’ability to understand and retain the concepts.Through a simple alteration
to the normal form of a game,students can clearly see the results of employing
mixed strategies,practise calculating expected payoffs,visualise the accompanying
‘indifference’in the payoffs,and recognise the mixed strategy Nash equilibrium in a
familiar way.
Figure 3:Including the optimal mixed strategies
25
4
5
12
10
10
10
16
15
8
13 3/4
10
13
13 3/5
13
8 4/5
13
10
CD
1⁄4 C,
3⁄4 D
A
B
1⁄5 A,
4⁄5 B
Player 1
Player 2Teaching Mixed Strategy Nash Equilibrium to Undergraduates
85
Appendix
Solving for the optimal mixed strategies illustrated in Figure 3 using calculus.
Matching Pennies Rock-Paper-Scissors
0
0
+1
–1
–1
+1
–1
+1
0
0
+1
–1
+1
–1
–1
+1
0
0
Rock Paper Scissors
Rock
Paper
Scissors
P
l
a
y
e
r
1
Player 2
–1
+1
+1
–1
+1
–1
–1
+1
Heads Tails
Heads
Tails
P
l
a
y
e
r
1
Player 2
4
3
4
1
4
1
16
4
*
0 4 16
) (
0 8 ) 1 ( 16 12 ) 1 ( 4
) (
8 ) 1 )( 1 ( 16 ) 1 ( 12 ) 1 ( 4 ) (
1
1
1
=
=
= =
= + − =
∂
Π ∂
= − − − − + =
∂
Π ∂
− − + − + − + = Π
D
C
q
q
p
E
q q q q
p
E
q p q p q p pq E
5
4
5
1
5
1
25
5
*
0 5 25
) (
0 15 ) 1 ( 10 ) 1 ( 5 25
) (
15 ) 1 )( 1 ( 10 ) 1 ( 5 ) 1 ( 25 ) (
2
2
2
=
=
= =
= − =
∂
Π ∂
= − − − + − =
∂
Π ∂
− − + − + − + = Π
B
A
p
p
q
E
p p p p
q
E
q p q p q p pq EInternational Review of Economics Education
86
Notes
1 Pindyck and Rubinfeld (2005) describe ‘matching pennies’as,‘each player chooses
heads or tails and the two players reveal their coins at the same time.If the coins
match (i.e.both are heads or both are tails),Player A wins and receives a dollar from
Player B.If the coins do not match,Player B wins and receives a dollar from Player A.’A
variant is for the winning player to receive the losing player’s coin.The normal forms
for the ‘matching pennies’and ‘rock-paper-scissors’games are given in the appendix.
2 These universities represent a (limited) cross-section of schools of higher education
across the US.While this selection of universities is limited to the US,note that we
have not found any textbook or article which presents the method we propose.
3 For example,player 2’s expected payoff from playing C is 
1⁄2*25 + 
1⁄2*10 = 17.5.Player
1’s expected payoff when player 2 plays C is 
1⁄2*4 + 
1⁄2*16 = 10.These are the expected
payoffs listed in the cell resulting from player 2 playing C and player 1 playing a mix
of 
1⁄2A and 
1⁄2B in the bottom left corner of Figure 2.
4 For a more thorough explanation about how to set up and derive the algebraic
solutions,please see Hay et al.;Nicholson and Snyder;McGuigan et al.;Neilson and
Winter;Pindyck and Rubinfeld;or Varian.We follow a fairly standard procedure when
solving for the equilibrium mixed strategies:Pik indicates the probability that player i
plays strategy k.Note that a player’s probability weights sum to one;i.e.P2C + P2D =
1.Therefore  P2D = 1 - P2C.Similarly,P1A + P1B = 1 and therefore P1B = 1 - P1A.
5 Although not directly related to the topic,we also ﬁnd that undergraduate students
are interested to learn that games in the normal,or strategic,form generally have a
ﬁnite and odd number of Nash equilibria (see Fudenberg and Tirole,1996).We
frequently have them solve a simple ‘Meeting Game’which results in 3 Nash
equilibria:2 in pure strategies and 1 in mixed strategies.A nice,easy example is on
page 247 in Mas-Colell et al.(1995).
References
Baldani,Jeffrey,James Bradﬁeld and Robert W.Turner (2005) Mathematical Economics
(2nd edition).Mason,OH:Thomson South-Western.
Bernheim,B.Douglas and Michael D.Whinston (2008) Microeconomics.New York,NY:
McGraw-Hill Irwin.
Chiappori,Pierre-Andre,Steven Levitt and Timothy Groseclose (2002) ‘Testing Mixed-
Strategy Equilibria when Players are Heterogeneous:The Case of Penalty Kicks in Soccer’,
American Economic Review.Volume 92,September.
Fudenberg,Drew and Jean Tirole (1996) Game Theory. Cambridge and London:The MIT
Press.
Hay,Frederick G.,Christine Oughton and Andrew S.Skinner (1996) Intermediate
Microeconomics:A Perspective on Price Theory.Manchester and New York:Manchester
University Press.
Mas-Colell,Andreu,Michael D.Whinston and Jerry R.Green (1995) Microeconomic Theory.
New York and Oxford:Oxford University Press.
McCain,Roger A.(2004) Game Theory:A Non-Technical Introduction to the Analysis of
Strategy.Mason,OH:Thomas South-Western.
McGuigan,James R.,R.Charles Moyer and Frederick H.de B.Harris (2005) Managerial
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87
Neilson,William S.and Harold Winter (1998) Intermediate Microeconomics.Cincinnati,OH:
South-Western College Publishing.
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Review.Volume 1,December.
Contact details
Evan Moore
Associate Professor and Head of the Department of Economics 
Auburn University Montgomery
P.O.Box 244023
Montgomery,AL 36124-4023
USA
Email: emoore1@aum.edu

Daniel L.Rubinfeld (2005) Microeconomics (6th edition).Upper Saddle River,NJ:Pearson

Game Theory. Cambridge and London:The

Intermediate Microeconomics and Its Application (10th edition).Mason,OH:Thomson South-Western.

Intermediate Microeconomics:A Perspective on Price Theory.Manchester and New York:Manchester

McCain,Roger A.(2004) Game Theory:A Non-Technical Introduction to the Analysis of Strategy.Mason,OH:Thomas South-Western.

McGuigan,James R.,R.Charles Moyer and Frederick H.de B.Harris (2005) Managerial Economics:Applications,Strategy,and Tactics.Mason,OH:Thomas South-Western.Teaching Mixed Strategy Nash Equilibrium to Undergraduates Neilson,William S.and Harold Winter

Microeconomic Theory. New York and Oxford:Oxford

Minimax Play at Wimbledon’,American Economic Review.Volume 1,December. Contact details Evan Moore Associate Professor and

Testing MixedStrategy Equilibria when Players are Heterogeneous:The Case of Penalty Kicks in Soccer’, American Economic Review.Volume 92,September.

Teaching Mixed Strategy Nash Equilibrium to Undergraduates

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