Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Game Theory
Giorgio [email protected]
https://mail.sssup.it/∼fagiolo/welcome.html
Academic Year 2005-2006
University of Verona
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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)
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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.
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Level-2 Rationality (2/3)
• 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.
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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!
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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)
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Level-3 Rationality (2/3)
• Three Crucial Questions:
– Why should players play a NE?
– Does a NE always exist?
– When a NE exists, is it always unique?
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Level-3 Rationality (2/3)
• 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”...
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Level-3 Rationality (3/3)
• Does a NE always exist? No, even in simplest games...
Player 2(π1, π2) H T
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)
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Level-3 Rationality (3/3)
• Does a NE always exist? No, even in simplest games...
Player 2(π1, π2) H T
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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.
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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.
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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)
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Classification of 2-person 2× 2 Symmetric Games
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
Classification of 2-person 2× 2 Symmetric Games
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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)]
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
II. Equilibrium Selection (2/3)
• 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∗.
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
II. Equilibrium Selection (3/3)
• 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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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
•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit
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 )