ECON7073: Microeconomic Analysis
Lectures Week 09 - Strategic Games
Simon Grant
3 - 6 October 2023
References
∗ Osborne (2.1-2.9, 3.1-3.2)
∗ Varian (29.1-29.2, 29.4, 30.1, 28.5-28.9)
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Games
∗ Game theory provides a natural extension of the single-agent
choice theory that we have discussed so far in the course.
∗ Game theory refers to the analysis of models of multiple
interacting decision-makers with potentially conflicting objectives.
∗ Game theory provides some of the principal modelling techniques
for most fields of economics, and has also yielded important
insights in many other disciplines, such as political science, law,
biology, and computer science.
∗ When constructing a game-theoretic model, we need to identify the
relevant aspects of a strategic situation, and then use general
principles that apply to various strategic settings to characterise
the possible outcomes.
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Models in game theory
∗ players
∗ available actions and order in which players move
(sequential vs. simultaneous)
∗ what is the players’ knowledge/information when they have to
make
a move (complete vs. incomplete, perfect vs. imperfect)
∗ what are the outcomes and how are they determined
by the joint actions
∗ what are the players’ preferences over outcomes/payoffs
resulting from outcomes
⇝ different classes of games and corresponding solution concepts
We will assume that all players in a game have preferences over
outcomes that can be represented by a utility (payoff) function.
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Solution concepts
A solution concept for a game attempts to characterise or predict what
outcomes might be expected when the game is played.
Solution concepts admit various possible interpretations:
(1) Descriptive (positive): How do players play in a given game?
(2) Prescriptive (normative): How “should” players play in a given
game?
(3) Theoretical: What outcomes are consistent with certain
assumptions regarding “reasonable” or “rational” behavior on the
part of the players?
⇝ (1) and (2)?
You should always keep the description/definition of a game
separate from the derivation of an associated solution!
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Strategic games (normal-form games)
Definition
A strategic game consists of
(1) a (finite) set of players,
with individual players denoted by i, j,. . . ,
(2) for each player i, a set of actions Ai, with a particular action of
player i denoted by ai,
(3) for each player i, preferences over the set of action profiles
A := ×iAi, which can be represented by a utility function
ui : A→ R.
Note:
∗ An action profile specifies a particular action for each player.
For example, in a two-player game, an action profile a must specify
an action a1 for player 1 and an action a2 for player 2,
so a = (a1, a2) ∈ A.
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∗ In a game with an arbitrary number of players we often denote the
actions of all players other than a given player i by a−i.
We can then write down an action profile a from the point of view
of player i as a = (ai, a−i).
∗ Since outcomes in a game are determined by the actions of all
players, i.e., by a given action profile, each player i’s utility is a
function of (ai, a−i), so we can write ui(ai, a−i).
Assumptions:
(i) Players choose actions simultaneously, without knowing the actions
chosen by other players.
(ii) The description of the game is known to all players.
(common knowledge!)
(iii) All players are rational (i.e., they have complete and transitive
preferences and aim to choose actions that yield most preferred
outcomes).
(iv) Players care only about instantaneous payoffs and not about future
effects of current actions (no inter-temporal strategic links).
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Examples
Pl. 2
C D
Pl. 1 C 2
2
3
−1
D
−1
3
0
0
Prisoner’s dilemma
B S
B
1
2
0
0
S
0
0
2
1
Battle of the Sexes
Stag Hare
Stag 2
2
0
−5
Hare −5
0
0
0
Stag hunt
H T
H
−1
1
1
−1
T
1
−1
−1
1
Matching Pennies
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Nash Equilibrium
Most commonly used solution concept for strategic games. It reflects a
stability or steady state requirement that no player has a strict
incentive
to unilaterally change her action given the actions of the other players
Definition
An action profile a∗ = (a∗i , a
∗
−i) is a Nash Equilibrium (NE)
if for each player i,
ui(a
∗
i , a
∗
−i) ⩾ ui(ai, a∗−i) for all ai ∈ Ai.
But players choose actions simultaneously, so how does a player know
what the other players will do?
Implicit requirements for Nash Equilibrium:
(1) No player has a strict incentive to unilaterally deviate given
(her beliefs about) the other players’ actions, and
(2) all players have correct beliefs about their opponents’ actions!Simon Gr nt (ANU) ECON7073 Strategic Games 8 / 28
Why should we assume that players know the actions of their
opponents?
(i) previous experience playing similar games (learning)
(ii) pre-play communication (a NE is a self-enforcing agreement)
(iii) social norms and conventions
(iv) focal points (“obvious” organising principles)
(v) evolution (only “high payoff” players survive in the long run)
How to find Nash equilibria?
Consider every action profile and check whether any player has a
profitable deviation.
Simon Grant (ANU) ECON7073 Strategic Games 9 / 28
Best response functions
Find the “best response” of a player for every fixed action profile of his
opponents. L C R
T
5
1
5
2
1
2
M
9
2
2
5
0
3
B
1
0
3
4
2
4
Definition
Player i’s best response function Bi : A−i → Ai is defined as
Bi(a−i) := {ai ∈ Ai |ui(ai, a−i) ⩾ ui(a′i, a−i) for all a′i ∈ Ai}.
The best response function is set-valued! It is a correspondence.
Proposition
An action profile a∗ is a NE iff (if and only if) a∗i ∈ Bi(a∗−i) for all i.
Simon Grant (ANU) ECON7073 Strategic Games 10 / 28
Guess 23 of the average
Each student announces an integer from 1 to 100, and a prize (say, $30)
is split equally between all students whose integer is closest to 23 of the
average of all chosen numbers.
∗ The game has a unique Nash equilibria (NE),
in which players all announce 1.
∗ In experiments, the winning number usually lies somewhere
between 10 and 30.
∗ What goes wrong?
– Some players have objectives that differ from maximizing their
chance of winning.
– Some players are not rational.
– Some players do not believe that their opponents are rational.
∗ . . .
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Implications regarding the interpretation of solution concepts
(1) Descriptive (positive): How do players play in a given game?
(2) Prescriptive (normative): How “should” players play in a given
game?
(3) Theoretical: What outcomes are consistent with certain
assumptions regarding “reasonable” or “rational” behavior on the
part of the players?
⇒ (1) and (2)?
We need to reconsider our assumptions!
∗ different preferences
∗ irrational players
∗ lack of common knowledge of rationality—players believe that some
opponents may be irrational (games with incomplete information)
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Revisiting rationality
Rationality assumption: Players have well-defined preferences and
choose actions that yield most-preferred outcomes.
For NE, we also need to assume that players have correct beliefs about
their opponents’ equilibrium actions. Can we derive a solution concept
that only assumes rationality?
C D
C
2
2
3
−1
D
−1
3
0
0
Prisoner’s dilemma
In the Prisoner’s Dilemma, assuming rational agents results in (D,D)
being played. We do not need to assume correct beliefs about
opponents’ actions!
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Another example
L C R
T
0
1
2
1
1
0
B
3
0
1
0
0
2
(1) If player 2 is rational, he will not choose R
(since C yields a higher payoff independently of player 1’s actions).
(2) If player 1 is rational and knows that player 2 is rational,
he will not choose B (because if we eliminate R for player 2,
T always yields a higher payoff than B).
(3) If player 2 is rational and knows that player 1 is rational and that
player 1 knows that player 2 is rational then player 2 will choose C
(since it yields a higher payoff when 1 plays T ).
⇒ We get a unique predicted outcome, (T,C).
Simon Grant (ANU) ECON7073 Strategic Games 14 / 28
Rationalizability
Rationalizability is a solution concept derived from only assuming
rationality and common knowledge of rationality (. . . ).
Unfortunately, it often yields only “vague” predictions.
Simon Grant (ANU) ECON7073 Strategic Games 15 / 28
Strictly dominated actions
Definition
An action ai ∈ Ai strictly dominates an action bi ∈ Ai if
ui(ai, a−i) > ui(bi, a−i) for all a−i ∈ A−i.
If the above inequalities hold, we say that bi is strictly dominated (by
ai).
Proposition
An action that is strictly dominated cannot be part of any NE profile.
Proof. ?
Proposition
If the process of successive iterated eliminations of strictly dominated
strategies yields a unique remaining action profile in a game, the
resulting profile is the unique NE of this game.
Simon Grant (ANU) ECON7073 Strategic Games 16 / 28
Weakly dominated actions
Definition
An action ai ∈ Ai weakly dominates an action bi ∈ Ai if
ui(ai, a−i) ⩾ ui(bi, a−i) for all a−i ∈ A−i, and
ui(ai, a−i) > ui(bi, a−i) for at least one a−i ∈ A−i.
L C R
T
2
2
4
4
3
1
B
3
2
1
3
0
0
T weakly dominates B, even though (B,L) is a NE!
Simon Grant (ANU) ECON7073 Strategic Games 17 / 28
Example: A voting game
∗ 3 citizens (the players) must vote for one of two candidates A or B.
∗ The candidate that obtains a majority of votes wins the election.
∗ Each citizen strictly prefers one of the two candidates.
Claim: Voting for less preferred candidate is weakly dominated by
voting for favorite candidate.
To see this, consider citizen i, and assume that A ≻i B. Check how i’s
vote affects the outcome for every strategy profile of his opponents, a−i
. . .
Each player voting for his preferred candidate is a NE.
There are other NE, e.g., where all players vote for candidate B!?
Simon Grant (ANU) ECON7073 Strategic Games 18 / 28
Cournot’s duopoly model (1838)
∗ players: two firms i ∈ {1, 2}
∗ actions: output choices qi ⩾ 0 (total output: Q = q1 + q2)
∗ preferences: maximise profits
πi(. . .) = P (Q)qi − Ci(qi),
where
P (Q) =
{
a− bQ, if Q ⩽ ab ,
0, if Q > ab ,
and Ci(qi) = cqi, with a , b , c > 0.
When q1 + q2 ⩽ a/b, firm i’s payoff function is given by
πi(q1, q2) = qi (a− b (q1 + q2))− cqi.
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To find firm 1’s best response function, we need to solve the
maximization problem
max
q1⩾0
{q1 (a− b (q1 + q2)− c)} .
Case 1: If q2 ⩾ (a− c)/b, b1(q2) = 0.
Case 2: If q2 < (a− c)/b, we can use the necessary condition to get
∂
∂q1
{−bq21 + (a− c− bq2)q1} = 0
⇒ −2bq1 + a− c− bq2 = 0
⇒ q1 = a− c− bq2
2b
=
a− c
2b
− 1
2
q2
Simon Grant (ANU) ECON7073 Strategic Games 20 / 28
We get the following (single-valued) best response functions:
B1(q2) =
{
a−c
2b − 12q2, if q2 < (a− c) /b,
0, if q2 ⩾ (a− c) /b,
B2(q1) =
{
a−c
2b − 12q1, if q1 < (a− c) /b,
0, if q1 ⩾ (a− c) /b,
Graph . . .
An output profile (q∗1, q∗2) is a NE iff q∗1 = B1(q∗2) and q∗2 = B2(q∗1).
Solving these two equations yields
q∗1 = q
∗
2 =
a− c
3b
.
Simon Grant (ANU) ECON7073 Strategic Games 21 / 28
Example: with linear (inverse)demand
and constant marginal cost
Recall for Perfect Competition and Monopoly
• Inverse demand p(Q) = 100− 2Q
• Constant marginal cost c = 20
• Perfect competition pc = c = 20
– Qc = 40 (since 100− 2× 40 = 20)
– CS = 12 (100− 20) 40 = 1600 and PS = 0
• Monopoly MR = c
– Qm = 20 (since 100− 4× 20 = 20)
– pm = 100− 2× 20 = 60
– CS = 12 (100− 60) 20 = 400 and PS = 40× 20 = 800
– Monopoly DWL = 400.
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Example cont.
Cournot-Nash equilibrium
q∗1 = q
∗
2 =
a− c
3b
=
100− 20
3× 2 =
40
3
So
QCN = q∗1 + q
∗
2 =
80
3
≈ 26.67
• Industry output in Cournot-Nash equilibrium greater than
monopoly output (20) but less than competitive output (40)
• Market price determined by inverse demand
pCN = 100− 2× 80
3
≈ 46.67
– Price in Cournot-Nash equiilibrium is less than in monopoly
but greater than in competitive equilibrium.
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Example cont.: comparision of DWLs
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Bertrand’s duopoly model (1883)
∗ players: two firms i ∈ {1, 2}
∗ actions: prices pi ⩾ 0 (real numbers ⇒ perfectly divisible)
∗ preferences: maximise profits
Assumptions:
(i) linear demand function:
D(p) =
{
a− p, if p ⩽ a
0, if p > a
(ii) Consumers only purchase the good from the firm that sets the
lowest price; if more than one firm sets the lowest price, all such
firms share the demand equally.
(iii) All firms can produce enough (at a unit cost c < a) to satisfy the
demand.
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We get the following profit functions:
πi(p1, p2) =

(pi − c)(a− pi), if pi < pj ,
1
2(pi − c)(a− pi), if pi = pj ,
0, if pi > pj .
Denote the monopoly price by pm, i.e., pm solves
max
p⩾0
{(p− c)(a− p)} ⇒ pm = a+ c
2
.
It is easy (?!) to see that no price profile other than (c, c) can be a NE.
But it is much harder to derive this result using best response functions:
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(1) pj < c: Bi(pj) = {pi | pi > pj}
(2) pj = c: Bi(pj) = {pi | pi ⩾ pj}
(3) c < pj ⩽ pm: Bi(pj) = ∅ (the empty set!)
(4) pj > pm: Bi(pj) = {pm}
Graph . . .⇒ unique NE (c, c)
If prices are discrete (i.e., in 1 cent increments),
there exists another NE (c+ 1, c+ 1).
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Stark Conclusion from Bertrand’s model
• Under price competition firms price at marginal cost.
– Does not seem very realistic prediction.
• Compare Cournot-Nash equilibrium which lies between perfect
competition and monopoly
– But firms are competing in quantities which also seems unrealistic.
Q. How might we proceed?
A.1 Capacity constraint - What if a firm is unable to serve entire
market?
– Consider 2-stage game:
1. In first stage, each firm (simultaneously) choose its capacity.
2. In second stage, firms compete in prices but a firm can only supply
the market up to the capacity it installed in the first stage.
A.2 Product differentiation
– you are asked to analyze such a model in Problem Set 5
– you will see by introducing product differentiation, the equilibrium
price lies between the marginal cost and the monopoly price.Simon Grant (ANU) ECON7073 Strategic Games 28 / 28