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Game Theory

Giorgio [email protected]

https://mail.sssup.it/∼fagiolo/welcome.html

Academic Year 2005-2006

University of Verona


Summary

1. Why Game Theory?

2. Cooperative vs. Noncooperative Games

3. Description of a Game

4. Rationality and Information Structure

5. Simultaneous-Move (SM) vs. Dynamic Games

6. Analysis

7. Examples

8. Problems and Suggested Solutions


Analysis

• Main Question

– What outcome should we expect to observe in a game played by fully rational players

with perfect recall and common knowledge about the structure of the game ?

• Answer

– It depends on whether:

∗ Agents know about others’ payoffs

∗ Rationality is common knowledge

∗ Players’ conjectures about each other’s play must be mutually correct

• Notation

– s = {si, s−i} = {si, (s1, ..., si−1, si+1, ..., sN)}

– S = S1 × S2 × · · · × SN

– S−i = S1 × S2 × · · · × Si−1 × Si+1 × · · · × SN

– Pure vs. Mixed Strategies


Level-1 Rationality

• Assume H1 only: Players are rational and know the structure of the game.

Definition 1 (Strictly dominant strategy) A strategy si ∈ Si is a strictly dominant strategy for playeri in game ΓN if for all s′i 6= si and s−i ∈ S−i

πi(si, s−i) > πi(s′i, s−i).

Definition 2 (Strictly dominated strategies) A strategy si ∈ Si is strictly dominated (SD) for playeri in game ΓN if there exists another strategy s′i ∈ Si such that for all s−i ∈ S−i

πi(s′i, s−i) > πi(si, s−i).

• A rational player satisfying H1 will:

– play a strictly dominant strategy if there exists one;

– not play a strictly dominated strategy.

• Problem: the outcome of the game is far from being unique

– It is rare that a strictly dominant strategy exists

– Many options resist to deletion of strictly dominated strategies


Level-1 Rationality: Examples

• A Strictly Dominant Strategy in the Prisoner Dilemma

Player 2(π1, π2) DC C

DC (−2,−2) (−10,−1)Player 1

C (−1,−10) (−5,−5)

• Deletion of a Strictly Dominated Strategy (D for Player 1)

Player 2(π1, π2) L R

U (+1,−1) (−1,+1)Player 1 M (−1,+1) (+1,−1)

D (−2,+5) (−3,+2)

• No strict dominance in Matching Pennies 2.0

Player 2(π1, π2) H T

Player 1 H (−1,+1) (+1,−1)T (+1,−1) (−1,+1)


Level-2 Rationality (1/3)

• Assume that H1, H2 and H3 hold: Players know others’ payoffs and that all are rational.

• Players know that others will not play strictly dominated strategies.

• Deletion of strictly dominated strategies can be iterated.

• Unique predictions can be sometimes reached.

• Example

Player 2

(π1, π2) DC C

DC (0,−2) (−10,−1)

Player 1

C (−1,−10) (−5,−5)

– C is no longer a dominant strategy for player 1.

– C is still a dominant strategy for player 2.

– Player 1 knows that 2 will not play DC and can eliminate it.

– Now player 1 knows that Player 2 will play C, to which C is a dominant strategy.



• Iterative deletion of strictly dominated strategies (IDSDS) does not depend on the order of

deletion.

• When N = 2 the set of strategies that resist to IDSDS is the “prediction of the game”.

• When N > 2 assumption H1-H3 allows one to delete even more outcomes.

Definition 3 (Best Response) In game ΓN a strategy si ∈ Si is a best response for player i to s−i if forall s′i ∈ Si

πi(si, s−i) > πi(s′i, s−i).

A strategy si is never a best response if there is no s−i for which si is a best response.

– A strictly dominated strategy is never a best response, but there may exist strategies that are never abest-response which are not strictly dominated.

– Iterative elimination of strategies that are never a best-response leads to the set of “rationalizablestrategies”, which is generally smaller than the set of strategies that resist IDSDS.

• Rationalizable strategies are those that one can expect to occur in a game played by rational

agents for which H1-H3 hold true.


Level-2 Rationality (3/3)• Example

Player 2

(π1, π2) l m n

U (5, 3) (0, 4) (3, 5)

Player 1 M (4, 0) (5, 5) (4, 0)

D (3, 5) (0, 4) (5, 3)

• Best-Response Strategies

s∗1 =

U

M

D

if

if

if

s2 = l

s2 = m

s2 = n

, s∗2 =

n

m

l

if

if

if

s1 = U

s1 = M

s1 = D

• Thus:

– No strategy is never a best response. Iterative elimination cannot be applied (Hint: with 2 players astrategy is never a BR ⇔ it is strictly dominated).

– All strategies can be rationalized by H1-H3 through a chain of justifications.

– Example for (U, l): J1={P1 justifies U by the belief that P2 will play l, which can be justified if P1 thinksthat P2 believes that P1 plays D}. J2={P1 justifies J1 by thinking that P2 thinks that P1 believes thatP2 plays l} ... and so on!

– Beliefs can be mutually wrong ! H1-H3 do not require them to be mutually consistent!


Level-3 Rationality (1/3)• Assume that H1-H3 hold and that players are mutually correct in their beliefs

Definition 4 (Nash Equilibrium (NE)) A strategy profile (s1, ..., sN ) is a Nash equilibrium for the gameΓN if for every i ∈ I and for all s′i ∈ Si

πi(si, s−i) > πi(s′i, s−i)

i.e. if each actual player’s strategy is a best-response.

• Examples:Player 2

(π1, π2) l m n

U (5, 3) (0, 4) (3, 5)Player 1 M (4, 0) (5,5) (4, 0)

D (3, 5) (0, 4) (5, 3)

Player 2(π1, π2) DC C

DC (−2,−2) (−10,−1)Player 1

C (−1,−10) (-5,-5)



• Three Crucial Questions:

– Why should players play a NE?

– Does a NE always exist?

– When a NE exists, is it always unique?



• Three Crucial Questions:

– Why should players play a NE?

– Does a NE always exist?

– When a NE exists, is it always unique?

• Why should players beliefs be mutually correct?

– Not a consequence of rationality

∗ NE as obvious ways to play the game

∗ NE as Pre-play commitments

∗ NE as Social Conventions

– No reasons to expect players to play NE are given in the rules of the game

– Need to complement the theory with “something else”...



• Does a NE always exist? No, even in simplest games...


Player 1 H (−1,+1) (+1,−1)T (+1,−1) (−1,+1)

– Existence is guaranteed in the space of mixed strategies for finite-strategy games

– Proof: See MWG p.250-253 (Hemicontinuous Corr, Kakutani Fixed Point Theorem)



• Does a NE always exist? No, even in simplest games...


Player 1 H (−1,+1) (+1,−1)T (+1,−1) (−1,+1)

– Existence is guaranteed in the space of mixed strategies for finite-strategy games

– Proof: See MWG p.250-253 (Hemicontinuous Corr, Kakutani Fixed Point Theorem)

• When a NE exists, is it always unique?

Player 2(π1, π2) A B

Player 1 A (1,1) (0, 0)B (0, 0) (2,2)

– Uniqueness and efficiency are not guaranteed

– When multiple NE arise, no selection principle is given


Summary

1. Why Game Theory?





6. Analysis

7. Examples



Some Examples

• Focus on 2-player 2× 2 symmetric games (πi = π all i ∈ I)

– Adding (heterogeneous) players and strategies only complicates the framework

– Games become more difficult to solve

– No new intuitions: problems always are existence and uniqueness

+1 −1

+1 a b

−1 c d

+1 −1

+1 1 0

−1 α β

where: a ≥ d, a > b and

α =c− b

a− b, β =

d− b

a− b

so that α ∈ R and β ≤ 1


Case I: Prisoners’ Dilemma Games

• Call +1=“Cooperate” and −1=“Defect”:

+1 −1

+1 a b

−1 c d

+1 −1

+1 1 0

−1 α β

PD : c > a > d > b.

PD : 0 < β < 1 and α > 1

+1 is strictly dominated by −1

(−1,−1) is the unique (inefficient) Nash


Case II: Coordination Games (CG)

• We have that:

+1 −1

+1 a b

−1 c d

+1 −1

+1 1 0

−1 α β

CG : a > c and d > b

CG : 0 < β ≤ 1 and α < 1

There are two NE: (+1, +1) and (−1,−1)

β < 1 : (+1, +1) is Pareto-efficient, (−1,−1) is Pareto-inferior

β = 1 : (+1, +1) and (−1,−1) are Pareto-equivalent

• A Pure-Coordination Game arises when α = 0.


Case IIa: CG with Strategic Complementarities

• A 2 person, 2 × 2 symmetric game is a game with strategic complementarities if the expected payofffrom playing +1 (resp. −1) is increasing in the probability that the opponent is playing +1 (resp. −1).

• Applications: Technological adoption, technological spillovers, etc..

• A 2 person, 2 × 2 symmetric game is a game with strategic complementarities if it is a coordinationgame and in addition:

SC : a > b, d > c

• SC games are therefore characterized by:

SC : − 1 ≤ α ≤ 1, 0 ≤ β ≤ 1 and − 1 ≤ α− β < 0

+1 −1

+1 1.0 0.0−1 0.5 0.8


Case IIb: Stag-Hunt Coordination Games

• A 2 person, 2 × 2 symmetric game is called a stag-hunt game if (i) (+1,+1) is Pareto dominant (a > d);and (ii) c > d.

SH : − 1 ≤ α ≤ 1, 0 ≤ β ≤ 1 and 0 ≤ α− β < 1

+1 −1

+1 1.0 0.0−1 0.5 0.3

• Applications: Pre-Play Commitments.

– Suppose that player I commits in a pre-play talk to play the efficient strategy +1. Would he be credible? No.

– Why ? Because player I cannot credibly communicate this intention to player II as it is always in playerI’s interest to convince II to play +1.

– Indeed, if player I is cheating, he is going to get always a larger payoff if player II will play +1, as c > d.

– Hence, by convincing II to play +1, player I will always get a gain and the pre-play commitment is notcredible. That is why pre-play communication does not ensure efficiency in a SH game.

– Conversely, in SCG, pre-play commitment is credible and can ensure efficiency.


Case III: Hawk-Dove Games

• A 2 person, 2× 2 symmetric game is called a hawk-dove game if:

HD : c > a > b > d

HD : α ≥ 1 and β ≤ 0

• Interpretation

– Strategy +1 is ’dove’ and strategy −1 is ’hawk’.

– There is a common resource of 2 units.

– If two +1 meet, they share equally and get a payoff of 1 each.

– If a +1 and a −1 meet, then +1 gets 0 and −1 gets all 2 units.

– If two −1 meet, then not only they destroy the resource, but they get a negative payoff of −1 each.

+1 −1

+1 1 0

−1 2 −1

There are two NE: (+1,−1) and (−1,+1)


Case IV: Efficient Dominant Strategy Games

• PD Games have an inefficient dominant strategy. EDS games have instead an efficient

dominant strategy:

EDS : a > c and b > d

EDS : α < 1 and β < 0

+1 −1

+1 1 0

−1 α β

−1 is strictly dominated by +1

(+1, +1) is the unique efficient NE


Classification of 2-person 2× 2 Symmetric Games


Summary

1. Why Game Theory?





6. Analysis

7. Examples



I. Incomplete Information (1/2)

• What happens if information is incomplete (nature moves first) and therefore imperfect (info

sets are not singletons)?

• Example: Harsanyi Setup

1. Nature chooses a type for each player

(θ1, ..., θN ) ∈ Θ = Θ1 × · · · ×ΘN

2. Joint probability distribution F (θ1, ..., θN ) common knowledge

3. Each player can only observe θi ∈ Θi

4. Payoffs to player i are:

πi(si, s−i; θi)

5. A pure strategy for i is a decision rule si(θi)

6. The set of pure strategies for player i is the set <i of all possible decision rules (i.e. functions si(θi))

7. Player i’s expected payoff is given by:

πi(s1(·), ..., sN (·)) = Eθ[πi(s1(θ1), ..., sN (θN ); θi)]


I. Incomplete Information (2/2)

• Bayesian Nash Equilibrium

Definition 5 A Bayesian Nash Equilibrium (BNE) for the game [I, {Si}, {πi(·)},Θ, F (·)] is a profile ofdecision rules (s1(·), ..., sN (·)) that is a NE for the game [I, {<i}, {πi(·)}].

• Amount of information and computational abilities required is huge!

– In a BNE each player must play a BR to the conditional distribution of his opponents’ strategies foreach type θi he can have !

– Each player is actually split in a (possibly infinite) number of identities, each one associated to anelement of Θi

– It is like the game were populated by a (possibly infinite) number of players

• Only very simple games can be handled

• Multiple players? Multiple Nature Moves?

• Commitment to full rationality and complete-graph interactions become too stringent


II. Equilibrium Selection (1/3)

• The theory is completely silent on:

– Why players should play a NE

– What happens when a NE does not exist

– What happens when more than one NE do exist (e.g., efficiency issues)

• Complementing NE theory with equilibrium refinements

– Additional criteria that (may) help in solving the equilibrium selection issue

– Similar to tatonnement in general equilibrium theory

– So many refinement theories that the issue shifted from equilibrium selection to equilibrium refinementsselection...

– Each refinement theory can be ad hoc justified

• Example: Trembling Hand Perfection (MWG, 8.F)

– Consider the mixed-strategy game ΓN = [I,∆(Si), πi]

– The mixed-strategy set ∆(Si) means that players choose a probability distribution over Si, i.e. play astrategy σi in the K-dim simplex

– A pure strategy is a vertex of the simplex

– The boundary of the simplex means playing some strategy with zero probability



• Example: Cont’d

– Problem: Some pure-strategy Nash equilibria are the result of an “excess of rationality”, i.e. they didnot occurred if the players knew how to move slightly away from them

– NE with weakly-dominated strategies

A B

A 4 3B 0 3

– Question: What if we force players to play every pure-strategy with a small but positive probability(make mistakes)?

– Players now must choose a strategy in the interior of the simplex

– More formally: Define εi(si) for all si ∈ Si and i ∈ I and allow players to choose mixed strategies s.t.the probability that player i plays the pure-strategy si is larger that εi(si).

– Define a perturbed-game ΓN,ε as the original game ΓN when a particular choice of lower-bounds εi(si)for all si ∈ Si and i ∈ I is made.

– A mixed-strategy NE σ∗ of the original game will be a Trembling-Hand Perfect (THP) NE if there existsa sequence of perturbed games that converges to the original game (as mistake sizes go to zero) whoseassociated NE stay arbitrarily close to σ∗.



• A THP NE always exists if the game admits a finite number of pure strategies

• Main Result: If σ∗ is a THP Nash equilibrium, then it does not involve playing weakly-

dominated strategies.

• If we decide not to accept equilibria that involve weakly-dominated strategies, and we are

dealing with games with a finite number of pure strategies, we are sure that at least a THP

NE does exist.

• Problems:

– Some important games have an infinite number of pure-strategies (continuous): Bertrand oligopolygames

– Difficult to find out THP NE if the game becomes more complicated (multiple players, incompleteinformation, etc.)

– Again: Commitment to full rationality and complete-graph interactions become too stringent


Concluding Remarks (1/2)

• Theory for simultaneous-move games very useful to answer very (too?) simple questions

(“low fat modeling”)

• Restrictions on analytical treatment imposed by rationality requirements and interaction

structure (all interact with everyone else) often become too stringent

• Games become very easily untractable and/or generate void implications when

– Many heterogeneous players

– Information is incomplete and/or imperfect

– Interaction structure not a complete graph

– Moves are not simultaneous: Dynamic games and “anything can happen” kind of results

• Need to go beyond full rationality paradigm, representative-individual philosophy and pecu-

liar interaction structures

• Relevant literature: Evolutionary-game theory and beyond


Concluding Remarks (2/2)• An alternative class of models

– Agents: i ∈ I = 1, 2, ..., N

– Actions: ai ∈ Ai = {ai1, ..., aiK}

– Time: t = 0, 1, 2, ...

– Dynamics:

∗ At each t (some or all) agents play a game Γ with players in Vi

∗ Interaction sets Vi ⊆ I define the interaction structure

∗ Agents are boundedly-rational and adaptive: they form expectations by observing actions played inthe past by their opponents

∗ Time-t payoffs to agent i are give by some function wt(a; at−1j , j ∈ Vi) where a ∈ Ai

∗ Players update their current action at each t e.g. by choosing their myopic BR to observed config-uration

ati ∈ arg max

a∈Ai

wt(a; at−1j , j ∈ Vi)

– Looking for absorbing states or statistical equilibria of the (Markov) process governing the evolution ofan action configuration (or some statistics thereof) as t →∞

at = (at1, ..., a

tN )

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