issues on the border of economics and computation נושאים בגבול כלכלה וחישוב
DESCRIPTION
Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב. INTRODUCTION. Instructors. Dr. Liad Blumrosen ד"ר ליעד בלומרוזן Department of economics, huji . Dr. Michael Schapira ד"ר מיכאל שפירא School of computer science and engineering, huji . - PowerPoint PPT PresentationTRANSCRIPT
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Issues on the border of economics and computation
וחישוב כלכלה בגבול נושאים
INTRODUCTION
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Instructors
• Dr. Liad Blumrosen "בלומרוזן ליעד ר ד– Department of economics, huji.
• Dr. Michael Schapira "שפירא מיכאל ר ד– School of computer science and engineering, huji.
• Office hours: by appointment.
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Course requirements• Attend (essentially all) classes.
• Solve 3-4 problem sets.– The final problem set might be slightly bigger.
• Problem sets grade is 100% of the final grade.– No exam, no home exam.
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Computer science and economics ?!?
Today:– Introduction and examples– Game theory 1.0.1.
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Classic computer science
What a single computer can compute?
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Classic Economics
Analyzing the interaction between humans, firms, etc.
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New computational environments
• Properties:– Large-scale systems, belong to various economic
entities.– Participants are individuals/firms with different goals.– Participants have private information.– Rapid changes in users behavior.
Electronic markets Information providers
Social networks P2P networks
Internet
Mobile and apps
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Algorithmic game theory• Which tools can we use for analyzing such
environments?
• Interactions between computers, owned by different economic entities and different goals.
• New tools should be developed: algorithmic game theory
• The theory borrows a lot from each field.
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What tools should we use?
“Classic” CSNot handling, eg:
Incentives Asymmetric
information Participation
constraints
Economics / Game theoryNot handling, eg:
Tractability Approximation Various objectives
Algorithmic Game Theory:+ Design & evaluate systems with selfish agents.+ Real need from the industry.
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Few examples
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Example 1: Single-Item Auctions
2nd-price auction• Buyers submit bids• Highest bid wins• Winner pays the 2nd-
highest bid
In which auction would you bid higher?How do people behave in such auctions?Which one earns greater revenue for the seller?
1st-price auction• Buyers submit bids• Highest bid wins• Winner pays his own bid
Say that you need to sell a single (indivisible) item to a set of bidders.How can you do that?
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Example 1: Single-Item Auctions
• Auctions are part of the mechanism design literature.
• Mechanism design: economists as engineers.Design markets with selfish agent to achieve some desired goals.– Relation to computer science is straightforward.– Once a niche field in economics, now mainstream.
See this year’s Nobel prize (+ 2007, 1994)
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Example 2: Sponsored-search auctions
Bla
Search results Advertisements
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Example 2: Sponsored-search auctions
A real system: A simple interface short response time robustness
Selfish parties: Google vs. Yahoo vs. MSN Users Advertisers
Economic challenges, eg:
Which auction to use? Private info – how much advertisers will pay? Click Fraud Attract new advertisers payments per impression/click/action
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Example 3: FCC spectrum auctions• Multi-billion dollar auctions.
• Preferences for bundles of frequencies (Combinatorial auctions): Consecutive geographic areas. Overlaps, already owned spectrum.
• Sophisticated bidders– At&t, Verizon, Google.– Again, asymmetric information.
• Bottleneck: communication.
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Example 4: selfish routing
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• Many cars try to minimize driving time.• All know the traffic congestion (גלגלצ, WAZE)
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Externalities and equilibria
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• Negative externalities: my driving time increases as more drivers take the same route.
• In “equilibrium”: no driver wants to change his chosen route.
• Or alternatively:– Equilibrium: for each driver, all routes have the same
driving time.• (Otherwise the driver will switch to another route…)
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Efficiency, equilibrium.
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• Our question: are equilibria socially efficient?– Would it be better for the society if someone told each
driver how to drive?
• We would like to compare:– The socially-efficient outcome.
• What would happen if a benevolent planner controlled traffic.– The equilibrium outcome.
• What happens in real life.
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Network 1
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• Socially efficient outcome: splitting traffic equally– expected driving time: ½*1+½*1/2=3/4 – Exercise: prove this is efficient.
• The only equilibrium: everyone use lower edge.– Otherwise, if someone chooses upper link, the cost in
the lower link is less than 1.– Expected cost: 1*1=1
C(n)=n
C(n)=1 (million)• c(n) – the cost (driving time) to
users when n users are using this road.
• Assume that a flow of 1 (million) users use this network.
S T
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Network 1
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• Conclusion:– Letting people choose paths incurs a cost– “price of anarchy”
• The immediate question: if we have a ratio of 75% for this small network, can it be much higher in more complex networks? Which networks?
C(n)=n
C(n)=1 (million)
S T
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Network 2
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• In equilibrium: half of the traffic uses upper routehalf uses lower route.
• Expected cost: ½*(1/2+1)+1/2*(1+1/2)=1.5
c(n)=n
c(n)=1
S T
c(n)=n
c(n)=1
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Network 3
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• The only equilibrium in this graph:everyone uses the svwt route.– Expected cost: 1+1=2
• Building new highways reduces social welfare!?
c(n)=n
c(n)=1
S T
v
W
c(n)=n
c(n)=1
c(n)=0
Now a new highway
was constructed!
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Braess’s Paradox
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• This example is known as the Braess’s Paradox:
sometimes destroying roads can be beneficial for society.
• The immediate question: how can we choose which roads to build or destroy?
c(n)=n
c(n)=1
S T
v
W
c(n)=n
c(n)=1
c(n)=0
Now a new highway
was constructed!
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Example 5: Internet Routing
Establish routes between the smaller networks that make up the Internet
Currently handled by the Border Gateway Protocol (BGP).
AT&T
Qwest
Comcast
Level3
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Why is Internet Routing Hard?
Not shortest-paths routing!!!
AT&T
Qwest
Comcast
Level3
My link to UUNET is for backup purposes only.
Load-balance myoutgoing traffic.
Always chooseshortest paths.
Avoid routes through AT&T if at all possible.
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BGP Dynamics
1 2
d2, I’m
available
1, my routeis 2d
1, I’m available
Prefer routes
through 2
Prefer routes
through 1
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Two Important Desiderata
• BGP safety
– Guaranteeing convergence to a stable routing state.
• Compliant behaviour.– Guaranteeing that nodes (ASes)
adhere to the protocol.
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• We saw examples for modern systems that raise many interesting questions in algorithmic game theory.
• Next:a quick introduction to game theory
• Outline:– What is a game?– Dominant strategy equilibrium– Nash equilibrium (pure and mixed)
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Game Theory• Game theory involves the study of strategic
situations
• Portrays complex strategic situations in a highly simplified and stylized setting– Strategic situations: my outcome depends not only
on my action, but also on the actions of the others.
• A central concept: rationality– A complex concept. Many definitions.– One possible definition:
Agents act to maximize their own utility subject to the information the have and the actions they can take.
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Applications• Economics
– Essentially everywhere• Business
– Pricing strategies, advertising, financial markets…• Computer science
– Analysis and design of large systems, internet, e-commerce.• Biology
– Evolution, signaling, …• Political Science
– Voting, social choice, fair division…• Law
– Resolutions of disputes, regulation, bargaining…• …
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Game Theory: Elements• All games have three elements
– players– strategies– payoffs
• Games may be cooperative or noncooperative– In this course, noncooperative games.
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• Let’s see some examples….
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Example 1: “chicken”Chicken!!!
Swerve Straight
Swerve 0, 0 -1, 1Straight 1, -1 -10,-10
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Example 2: Prisoner’s Dilemma• Two suspects for a crime can:
– Cooperate (stay silent, deny crime).• If both cooperate, 1 year in jail.
– Defect (confess).• If both defect, 3 years (reduced since they confessed).
– If A defects (blames the other), and B cooperate (silent) then A is free, and B serves a long sentence.
Cooperate Defect
Cooperate -1, -1 -5, 0Defect 0, -5 -3,-3
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Lecture Outline• What is a game?
– Few examples.
Best responses
• Dominant strategies
• Nash Equilibrium– Pure– Mixed
• Existence and computation
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Notation• We will denote a game G between two
players (A and B) by
G[ SA, SB, UA(a,b), UB(a,b)]
whereSA = set of strategies for player A (a SA)SB = set of strategies for player B (b SB)UA : SA x SB R (utility function for player A)UB : SA x SB R utility function for player B
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Normal-form game: Example
• Example:– Actions:
SA = {“C”,”D”}SB = {“C”,”D}
– Payoffs:uA(C,C) = -1, uA(C,D) = -5, uA(D,C) = 0, uA(D,D) = -3
Cooperate Defect
Cooperate -1, -1 -5, 0Defect 0, -5 -3,-3
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A best response: intuition• Can we predict how players behave in a game?
First step, what will players do when they know the strategy of the other players?
• Intuitively: players will best-respond to the strategies of their opponents.
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A best response: Definition• When player B plays b. A strategy a* is a best
response to b if
UA(a*,b) UA(a’,b) for all a’ SA
(given that B plays b, no strategy gains A a higher payoff than a*)
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A best response: example
Example:When row player plays Up,what is the best response of the column player?
Left Right
Up 1,1 0,0Bottom 0,0 1,1
Left Right
Up 1,1 0,0Bottom 0,0 1,1
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Dominant Strategies ( / דומיננטיות שולטות (אסטרטגיות
• Definition: action a* is a dominant strategy for player A if it is a best response to every action b of B.
Namely, for every strategy b of B we have:
UA(a*,b) UA(a’,b) for all a’ SA
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Dominant Strategies: in the prisoner’s dilemma
Cooperate Defect
Cooperate -1, -1 -5, 0
Defect 0, -5 -3,-3
• For each player: “Defect” is a best response to both “Cooperate” and “Defect.
• Here, “Defect” is a dominant strategy for both players…
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• In the prisoner’s dilemma: (Defect, Defect) is a dominant-strategy equilibrium.
Dominant Strategy equilibriumשולטות באסטרטגיות משקל שווי
• Definition: (a,b) is a dominant-strategy equilibrium if a is dominant for A and b is dominant for B.– (similar definition for more players)
Cooperate Defect
Cooperate -1, -1 -5, 0Defect 0, -5 -3,-3
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Dominant strategies: another example
• Who has a dominant strategy in this game?
• Dominant-strategy equilibrium?
Left middle Right
Up 7,2 2,2 0,0Bottom 3,4 5,2 0,4
We allowed ≥ in the
definition. “Weakly
dominant”
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Dominant strategies: pros and cons
• Plus: Strong solution. – Why should I play anything else if I have a
dominant strategy?
• Main problem:Does not exist in many games….
Left Right
Up 1,1 0,0Bottom 0,0 1,1
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Lecture Outline• What is a game?
– Few examples.
• Best responses
• Dominant strategies (golden balls)
Nash Equilibrium– Pure– Mixed
• Existence and computation
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Nash Equilibrium• How will players play when dominant-strategy
equilibrium does not exist?– We will define a weaker equilibrium concept: Nash
equilibrium
• A pair of strategies (a*,b*) is defined to be a Nash equilibrium if:a* is player A’s best response to b*, and b* is player B’s best response to a*.
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Nash Equilibrium: Definition• A direct definition:
A pair of strategies (a*,b*) is defined to be a Nash equilibrium if
UA(a*,b*) UA(a’,b*) for all a’ SA
UB(a*,b*) Ub(a*,b’) for all b’ SB
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Nash Eq.: Interpretation• No regret: Even if one player reveals his strategy,
the other player cannot benefit.– this is not the case with non-equilibrium strategies
• Stability: Once we reach a Nash equilibrium, players have no incentive to alter their strategies.– Even after observing the strategies of the other players
• Necessary condition for an outcome chosen by rational players.– If players think that there is obvious outcome to the
game, it must be a Nash equilibrium
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(Pure) Nash Equilibrium• Examples:
Left Right
Up 1,1 0,0Bottom 0,0 1,1
Swerve Straight
Swerve 0, 0 -1, 1Straight 1, -1 -10,-10
Note: when column player plays “straight”, then “straight” is no longer a best response to the row player.
Here, communication between players help.
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Nash vs. Dominant Strategies• Every dominant strategy equilibrium is a Nash
equilibrium.– If a strategy is a best response to all strategies of
the other players, it is of course a best response to the dominant strategy of the other.
• The opposite is not true.
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Nash equilibrium: existence• Does a Nash equilibrium always exist?
– Note:we already saw that multiple equilibria are possible.
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Example 4:
• No (pure) Nash equilibrium.
• But how do people play this game?
-1,1 1,-1
1,-1 -1,1
Tail Heads
Tail
Heads
Matching Pennies ( פרט או (זוגIs this an
equilibrium?
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“Pure” Nash: pros and cons• Good:
– Describes “stable” outcomes.– May exist when dominant-strategy equilibria
does not exist.– Simple and intuitive (especially when unique).
• Bad:– Not unique.
• What happens when multiple equilibria exist?– Does not always exist!
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Mixed strategies• Consider the following strategy:
“I will toss a coin. With probability ½ I will choose bottom.With probability ½ I will choose up.”
• If lottery is allowed, now each player has an infinite number of strategies…
Left Right
Up 1,1 0,0Bottom 0,0 1,1
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Mixed strategies: Definition• Definition: a “mixed strategy” is a probability
distribution over actions.– If {a1,a2,…,am} are the pure strategies of A,
then {p1,…,pm} is a mixed strategy for A if
-1,1 1,-11,-1 -1,1
Tail Heads
Tail
Heads
m
iip
1
1
1/2
1/2
1/3
2/3
9/10
1/10
0
1
1/4
1/2
0ip
(1)
(2) For all i
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Pure and Mixed strategies• Clearly, every pure strategy is a mixed strategy
as well.– That gives probability 1 to one of the pure strategies.
• We will simply use the term “strategies”.
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Expected payoff• When the two players play mixed strategies, the
payoff is the expected payoff. (הממוצע)
L R
T 3, -1 4, 2
D 6, -5 1,9
2/3
1/3
3/41/4
• What is the payoff of the row player? when the players play sA=(2/3, 1/3) and sB=(1/4,3/4)
uA(sA,sB) = 2/3 * ¼ * 3 + 1/3 * ¼ * 6 + 2/3 * ¾ * 4 + 1/3 * ¾ * 1 = 3.25
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Best response (w. mixed strategies)• Definition:
Consider a mixed strategy sB of player B.
A strategy s* for player A is a best response to sB if no other pure strategy gains A higher expected payoff.
Namely,
– Note: we will later see that this implies that no mixed strategy is better for A than s*.
UA(s*,sB) UA(a’,sB) for all a’ in SA
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Best response (w. mixed strategies)
-1,1 1,-11,-1 -1,1
זוג פרטזוגפרט
3/41/4What is a best response to (1/4,3/4)?
What would you do if you knew that your opponent plays one strategy more frequently?
Will you play pure or mixed?
1
0
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Mixed strategies are realistic?• Do people randomize?
– Computers? Evolution? Stock markets? Teacher choosing questions in exams.
• Model long term behavior…• Model uncertainty about the other players.
• פרט או זוג• Basketball• Soccer
– How would you define strategy in penalty kicks?
– “the player that kicks more often to the left”
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Nash eq. with mixed strategies• Main idea: given a fixed behavior of the others, I
will not change my strategy.
• Definition: (SA,SB) are in Nash Equilibrium, if each strategy is a best response to the other.
-1,1 1,-11,-1 -1,1
זוג פרטזוגפרט
1/21/2
1/2
1/2
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Example: Battle of the SexesEquilibria in “battle
of the sexes”:
– Two pure equilibria.
– One mixed (2/3,1/3),(1/3,2/3)
2,1 0,0
0,0 1,2
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Lecture Outline• What is a game?
– Few examples.
• Best responses
• Dominant strategies
• Nash Equilibrium– Pure– Mixed
Existence and computation
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Existence of equilibria• Dominant strategies equilibria do not exist
in every game.– Same goes for Pure Nash equilibria.
• What about Nash equilibria (with mixed strategies)?
Good news: always exist.
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Nash’s Theorem• Theorem (Nash, 1950):
every game has at least one Nash equilibrium!– With some technical details about the set of strategies.– Proof uses fix-point theorems.
• Nash was awarded the Nobel prize for this work in 1994.
Nash equilibrium (with mixed strategies):– Good: always exists. Models long term stability.– Bad: Less simple and intuitive. Multiple equilibria exist.
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Computing Equilibria• Dominant strategy: for each player, check if she
has a dominant strategy.
• Pure Nash: for each combination of actions, check if a player has a beneficial deviation.
• How can we find Nash equilibria in general?– This is a real problem in large games.
• Area of extensive research.– Easy in “small” games.
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Summary• We learned about simultaneous-action games,
represented by a matrix of payoffs. (Games in their “normal form”)– Next topic: sequential games.
• We wanted to predict the steady/stable state behavior on the games, and defined concepts of equilibria.
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Finding mixed equilibria• We will use the following lemma:
Lemma: let sA be a best response to sB.If sA chooses the pure strategies a, a’ with positive probability, then
uA(a,sB)=uA(a’,sB)
Namely, if we sometime choose a and sometime choose a’, they gain us the same expected payoff (given a fixed behavior of the others).
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Proof of Lemma• Assume that in best response:
a is chosen with probability pa
a’ is chosen with probability pa’
– pa ,pa’ >0
• Now if uA(a,sB) > uA(a’,sB), then this is not a best response:– The same strategy that chooses
a with probability pa+pa’ and a’ with probability 0 gains A higher payoff.
pa’
pa
0
pa + pa’
sB
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Finding mixed equilibriaM S
M 2,1 0,0S 0,0 1,2
1/43/4• What is the best response to sB=(3/4,1/4)?
• Can it be sA=(½, ½)?uA(M,sB) = ¾*2 + 1/4*0 = 1.5uA(S,sB) = ¾*0 + 1/4*1 = ¼Expected payoff: ½*1.5 +
½*1/4(1/2,1/2) cannot be a best response, niether (0.99,0.01)
• adding more mass to M will increase expected payoff of A. • Again, here the best response is a pure strategy (“M”),
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Finding mixed equilibria• So how can we find (strictly) mixed-strategy
equilibria?
• We will use the lemma that we proved: if in equilibrium a player plays two pure strategies with positive probability, then the expected payoff from both strategies should be the same.
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Finding mixed equilibriaM S
M 2,1 0,0S 0,0 1,2
1-qq• Consider an equilibrium
sA=(p,1-p), sB=(q,1-q) (q,p>0)
then: uA(“M”,sB) = uA(“S”,sB) =
• If sA is a best response, we must have:uA(“M”,sB)=uA(“S”,sB)
that is : 2q = (1-q) q=1/3
• Similarly, if sB is a best response then p=2/3.
1-p
p
q*2 + 0*(1-q)q*0 + (1-q)*1
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Finding mixed equilibriaM S
M 2,1 0,0S 0,0 1,2
2/31/3• Note that since
all mixed strategies are best response to sB=(1/3,2/3).
But only sA=(2/3,1/3) ensures that sB=(1/3,2/3) is also a best response to sA.
1/3
2/3
uA(“M”,sB)=uA(“S”,sB)
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Equilibria• All we said extends to more players:
• (s1,…,sn) is a Nash equilibrium, if for every i, si is a best-response to the other strategies.
• (s1,…,sn) is a dominant-strategy equilibrium, if for every i, si is the best response to any other set of strategies.
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EquilibriaTake home message:
• Dominant-strategy equilibrium:
my strategy is the best no matter what the others do.
Exists in some games.
• Nash equilibrium:
my strategy is the best given what the others are currently doing.
Always exists.
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Example 1: coordination games
Left Right
Left 1,1 0,0Right 0,0 1,1
Row playerהשורות שחקן
Column Playerהעמודות שחקן
Right number: utility for
Row Player
Left number: utility for Column Player
Without laws, when this game is repeated, what will happen?