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ECON7073: Microeconomic Analysis

创建日期：2024-06-05 14:12:13

所属分类：经济Economics

标签： ECON7073

Microeconomic Analysis

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ECON7073: Microeconomic Analysis

Lectures Week 09 - Strategic Games
References
∗ Osborne (2.1-2.9, 3.1-3.2)
∗ Varian (29.1-29.2, 29.4, 30.1, 28.5-28.9)

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.

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.

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!

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.

∗ 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).

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

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’
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.

Best response functions
Find the “best response” of a player for every fixed action profile of his
opponents.
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.

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.
∗ . . .

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)

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?

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!

Another example

(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).

Rationalizability
Rationalizability is a solution concept derived from only assuming
rationality and common knowledge of rationality (. . . ).
Unfortunately, it often yields only “vague” predictions.
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.

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.

T weakly dominates B, even though (B,L) is a NE!

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!?

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.

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

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
.

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.

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.

Example cont.: comparision of DWLs

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.

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:

(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).

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.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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