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Applications of Rationalizability
ā¢ Location: Two competitors choose locations along a spectrum (politicians along left-right; vendors along beach; cereal-makers along sweetness)
ā¢ Partnership: Partners in enterprise individually choose effort,
which benefits enterprise but entails personal costwhich benefits enterprise but entails personal cost
ā¢ Social unrest: People choose whether to protest, according to
how zealous they are & how many protesters they expect
ā¢ Which strategy profiles are rationalizable? Nice insights!
1-2Ā© Sergio Turner
Where should vendors locate?
ā¢ Ipanema Beach, Rio de Janeiro, 1/3/14. Heat wave compels residents to the beach.
Where should vendors of coconut water locate?
1-3Ā© Sergio Turner
Location
ā¢ Players: Pat & Chris are to sell soda along a beach
Beach is divided into 9 regions each with 50 customers:
ā¢ Strategy sets: Each chooses location of his/her soda stand.ā¢ Strategy sets: Each chooses location of his/her soda stand.
Si = {1,2,ā¦,9}
ā¢ Payoffs: ui = 12.5 x R + possibly 6.25,
R := number of regions nearer iās location than otherās
They split the region, if any, that is equally near
1-4Ā© Sergio Turner
Location
ā¢ For examples, uC(2,5) = 3 x 12.5 = 37.50,
uP(2,5) = 6 x 12.5 = 75
ā¢ uC(1,9) = 4 x 12.5 + 6.25 = 56.25
uP(1,9) = ā¦symmetricā¦ = 56.25
ā¢ Could build a 9x9 matrix with 81 entries uC, uP ā¦
unnecessary, point of matrix is to help, not to burden
1-5Ā© Sergio Turner
Location ā iterated dominance
ā¢ Claim: For either, location 1 is dominated by location 2
ā¢ Let us show this for Chris, other case following by symmetry.
Show uC(1,y) < uC(2,y) for every location y Pat chooses
- Y = 1: uC(1,1) = (12.5 x 9)/2 = 56.25; uC(2,1) = 12.5 x 8 = 100
Ok, ā<ā!Ok, ā<ā!
- Y = 2: uC(1,2) = 12.5; uC(2,2) = 56.25
Ok, ā<ā!
- Y = 3: uC(1,3) = 12.5 + 6.25 = 18.75; uC(2,3) = 25
Ok, ā<ā!
- Etc: Chris, by locating in 2 vs 1, captures at least 1 extra region
away from Patās y>3, while keeping those captured under 1.
1-6Ā© Sergio Turner
Location ā iterated dominance
ā¢ Claim: For either, 9 is dominated by 8
True by symmetry. Reasoning like in prior claim, but from the
beachās other end instead.
ā¢ Seen that, for either, strats. 1 & 9 are dominated.ā¢ Seen that, for either, strats. 1 & 9 are dominated.
ā¢ Claim: No other strategies are dominated
ā¢ Proof: Suffices that other strats are b.r. to some belief (UD=B)
Chrisā 2 is not dominated b/c it is b.r. to belief Pat plays 1
Chrisā 3 is not dominated b/c it is b.r. to belief Pat plays 2 ā¦.
Chrisā 8 is not dominated b/c it is b.r. to belief Pat plays 9
1-7Ā© Sergio Turner
Location - rationalizability
ā¢ Conclude that dominated strategies are beach ends for either
ā¢ Deleting these gives game w/ reduced strategies R1i = {2,ā¦.,8}
ā¢ In this reduced game, by symmetry, dominated strategies are
beach ends, so deleting these reduces game to R2i = {3,ā¦.,7}
ā¢ ā¦ R3 = {4,5,6}, R4 = {5} ā no further reduction possibleā¢ ā¦ R3i = {4,5,6}, R4
i = {5} ā no further reduction possible
ā¢ So (sC,sP)=(5,5) is (the sole) profile that survives iterated
deletion of dominated strategies (that is rationalizable)
ā¢ āSolution conceptā of rationalizability predicts/prescribes
competitors along spectrum both locate in middle.
1-8Ā© Sergio Turner
Location - discussion
ā¢ Does this prediction conform to what we observe two
competitors do when they compete along spectrum?
ā¢ Think of Republican vs Democrat in presidential election.
They often do seem to āmove to the middleā upon having They often do seem to āmove to the middleā upon having
secured their partiesā nomination for the election. Wish to
steal away āswing votersā whoād vote for competitor o.w.
ā¢ Think of Pepsi and Coke.
Whether spectrum is color, fizziness, sweetness, size, they
have chosen the same location ā virtually identical sodas.
1-9Ā© Sergio Turner
Location - limitations
ā¢ What if there are more than two competitors?
ā¢ What if there are multiple dimensions, not just one spectrum?
E.g. cereals. Makers do not all market āmiddle cereal,ā there is great variety in
many dimensions: price, vitamins, sweetness, crunchiness, colors, size, ā¦
ā¢ What if a player can wait to observe the otherās location?ā¢ What if a player can wait to observe the otherās location?
ā¢ Important not to extrapolate conclusions here to settings that
are not alike
1-10Ā© Sergio Turner
Partnership
ā¢ Players: Two partners working on a joint project
ā¢ Strategies: Each player chooses a level of effort, Si = [0,4].
ā¢ Payoffs: ui = profit/2 ā cost of effort
profit := 4(x+y+cxy) where x,y are efforts & 0<c< Ā¼
cost of effort := square of effortcost of effort := square of effort
u1(x,y) = 2(x+y+cāxy) ā x2
u2(x,y) = 2(x+y+cāxy) ā y2
Each player raises profit by raising own effort
Complementarity: oneās effort made more effective by otherās
1-11Ā© Sergio Turner
Partnership ā best responses
ā¢ For a belief of player 1 about 2ās effort yĪµ[0,4], let y* = expecn
ā¢ 1ās payoff 2(x+y+cxy) ā x2 has expectation 2(x+y*+cxy*) ā x2
Strategy x is a b.r. to this belief if it maximizes this expecn
From calculus (take derivative wrt x, set to zero, solve for x):
ā¢ Note, BR function of belief only through the expecn y*.
See that if 1 expects 2 to exert higher effort, 1ās b.r. is higher
ā¢ By symmetry,
1-12
* *
1(y ) 1BR x cy= = +
* *
2 ( ) 1BR x y cx= = +
Ā© Sergio Turner
Partnership ā best responses
ā¢ BRi(e) = 1 +ce
Since eĪµ[0,4], BRi(e)Īµ[1,1+4c]
So Bi = [1,1+4c]
ā¢ Player j knows iās strats offā¢ Player j knows iās strats off
[1,1+4c] are dominated
ā¢ So reduced strategy space is
R1i = [1,1+4c] (recall c< Ā¼ )
1-13Ā© Sergio Turner
Partnership ā rationalizability
ā¢ Applying BRi(e) = 1 +ce in iterations ā¦
ā¦ to original [0,4] got R1i = [1,1+4c]
ā¦ to R1i , get R2
i = [1+c,1+c+4c2]
ā¦ to R2i , get R2
i = [1+c+c2,1+c+c2+4c3]
ā¢ Since |c|<1, these series converge to 1/(1-c) & 4c^(k+1)->0
ā¢ Conclude: (x,y)=(1/(1-c), 1/(1-c)) sole rationalizable profile
ā¢ Note: This also intersection of BRās: solve y=1+cx, x=1+cy.
1-14
1
1,..., 1,...,
, 4k i i k
i
i k i k
R c c c+
= =
= + ā ā
Ā© Sergio Turner
Partnership - discussion
ā¢ Rationalizability concept predicts both exert effort 1/(1-c)
ā¢ Profit and individual payoffs are
2 2
2 8 44( ) 4
1 (1 ) (1 )
4 2 1 3 2
c cx y cxy
c c c
c c
ā+ + = + =
ā ā ā
ā ā
ā¢ Is this efficient? No. Maximizing joint payoff profit āx2 ā y2:
ā¢ Efficiency requires more effort than rationalizable. Individuals
fail to internalize that own effort makes otherās more effective
ā¢ Inefficiency increases in c, extent of this complementarity. 1-15
2 2 2
4 2 1 3 2
(1 ) (1 ) (1 )
c cu
c c c
ā ā= ā =
ā ā ā
2 1
1 2 1x y
c c= = >
ā ā
Ā© Sergio Turner
Should I join protests?
Pro-Assad rally in 2011, Anti-Mubarak protest in 2011
Some more than others like to protest.
And anyone who must protest prefers big company.
1-16Ā© Sergio Turner
Social unrest
ā¢ Players: Uniformly distributed in interval [0,1]
Near 0 are āapathetic,ā near 1 are āzealotsā
ā¢ Strategies: For each player iĪµ[0,1], Si ={stay home H, protest P}
ā¢ Payoffs: Let x = fraction of people who choose to protest
ui(H,x) = 4x-2
ui(P,x) = 2(4x-2) + Ī±i
ā¢ Ī±>0 parameterizes how intensely protesting raises payoff
Note, zealots enjoy protesting more. Protesting worse if low x
1-17Ā© Sergio Turner
Social unrest ā best response
ā¢ Zealotry induces protesting, threshold falls with belief x & Ī±
ā¢ Will a protest occur? Turns out, answer hinges on whether
protesting is a b.r. for top zealot i=1 when she believes none
THRESHOLDxixHuxPu ii =āā„āā„ā Ī±/)42(),(),(b.r a is P
protesting is a b.r. for top zealot i=1 when she believes none
will (x=0), i.e on whether Ī±>2.
ā¢ Case Ī±=1. If player i believes most stay home (x< Ā¼), then
T>1>=i, so iās b.r. is H. Conversely, if x>Ā½, then T<0<=i, so iās
b.r. is P. So for all players both H,P are b.r., i.e. Bi = {H,P}, and
any profile is rationalizable.
ā¢ Low Ī± (protesting not so exciting) & strategic uncertainty
cause no protest, even if all better off protesting. 1-18Ā© Sergio Turner
Social unrest ā rationalizability
ā¢ Case Ī±=3.
ā¢ Consider i> 2/3. Then whatever x, T<=2/3 & iās b.r. is P: Bi={P}
Consider i<2/3. Depending on x, b.r. is either, so Bi={P,H}
ā¢ Delete H from all i>2/3. KNR implies all believe x>1/3.
THRESHOLDxixHuxPu ii =āā„āā„ā Ī±/)42(),(),(b.r a is P
ā¢ Delete H from all i>2/3. KNR implies all believe x>1/3.
But then T ā¤2/9, so P becomes dominant for players i>2/9
ā¢ Delete H from all i>2/9. KNR implies all believe x>7/9.
But then T = -10/27, so b.r. is P for all players i
ā¢ Sole rationalizable profile is āall protestā
ā¢ Conclude: High Ī± makes social unrest sole rationalizable profl
1-19Ā© Sergio Turner
Social unrest - discussion
ā¢ If Ī± is low (<2 actually), then any profile is rationalizable,
including for all to stay home.
Protestingās benefit, if any, comes from belief x and not own
type, and combo of low beliefs and inaction is ārationalā
ā¢ If Ī± is high (>2 actually), then āall protestā is sole ā¢ If Ī± is high (>2 actually), then āall protestā is sole
rationalizable profile: social unrest guaranteed.
ā¢ Factors that raise Ī±: Culture sympathetic to protesters, news
coverage of protesters fair/positive, police respectful, mayor
announces support, ā¦.
Protest organizers often do work on these dimensions.
1-20Ā© Sergio Turner