CHAPTER 8 Location, Partnership & Social Unrest

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CHAPTER 8

Location, Partnership & Social Unrest

1-1© Sergio Turner

Applications of Rationalizability

• Location: Two competitors choose locations along a spectrum (politicians along left-right; vendors along beach; cereal-makers along sweetness)

• Partnership: Partners in enterprise individually choose effort,

which benefits enterprise but entails personal costwhich benefits enterprise but entails personal cost

• Social unrest: People choose whether to protest, according to

how zealous they are & how many protesters they expect

• Which strategy profiles are rationalizable? Nice insights!

1-2© Sergio Turner

Where should vendors locate?

• Ipanema Beach, Rio de Janeiro, 1/3/14. Heat wave compels residents to the beach.

Where should vendors of coconut water locate?

1-3© Sergio Turner

Location

• Players: Pat & Chris are to sell soda along a beach

Beach is divided into 9 regions each with 50 customers:

• Strategy sets: Each chooses location of his/her soda stand.• Strategy sets: Each chooses location of his/her soda stand.

Si = {1,2,…,9}

• Payoffs: ui = 12.5 x R + possibly 6.25,

R := number of regions nearer i’s location than other’s

They split the region, if any, that is equally near

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Location

• For examples, uC(2,5) = 3 x 12.5 = 37.50,

uP(2,5) = 6 x 12.5 = 75

• uC(1,9) = 4 x 12.5 + 6.25 = 56.25

uP(1,9) = …symmetric… = 56.25

• Could build a 9x9 matrix with 81 entries uC, uP …

unnecessary, point of matrix is to help, not to burden

1-5© Sergio Turner

Location – iterated dominance

• Claim: For either, location 1 is dominated by location 2

• 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, “<“!Ok, “<“!

- Y = 2: uC(1,2) = 12.5; uC(2,2) = 56.25

Ok, “<“!

- Y = 3: uC(1,3) = 12.5 + 6.25 = 18.75; uC(2,3) = 25

Ok, “<“!

- Etc: Chris, by locating in 2 vs 1, captures at least 1 extra region

away from Pat’s y>3, while keeping those captured under 1.

1-6© Sergio Turner

Location – iterated dominance

• Claim: For either, 9 is dominated by 8

True by symmetry. Reasoning like in prior claim, but from the

beach’s other end instead.

• Seen that, for either, strats. 1 & 9 are dominated.• Seen that, for either, strats. 1 & 9 are dominated.

• Claim: No other strategies are dominated

• Proof: Suffices that other strats are b.r. to some belief (UD=B)

Chris’ 2 is not dominated b/c it is b.r. to belief Pat plays 1

Chris’ 3 is not dominated b/c it is b.r. to belief Pat plays 2 ….

Chris’ 8 is not dominated b/c it is b.r. to belief Pat plays 9

1-7© Sergio Turner

Location - rationalizability

• Conclude that dominated strategies are beach ends for either

• Deleting these gives game w/ reduced strategies R1i = {2,….,8}

• In this reduced game, by symmetry, dominated strategies are

beach ends, so deleting these reduces game to R2i = {3,….,7}

• … R3 = {4,5,6}, R4 = {5} – no further reduction possible• … R3i = {4,5,6}, R4

i = {5} – no further reduction possible

• So (sC,sP)=(5,5) is (the sole) profile that survives iterated

deletion of dominated strategies (that is rationalizable)

• “Solution concept” of rationalizability predicts/prescribes

competitors along spectrum both locate in middle.

1-8© Sergio Turner

Location - discussion

• Does this prediction conform to what we observe two

competitors do when they compete along spectrum?

• Think of Republican vs Democrat in presidential election.

They often do seem to “move to the middle” upon having They often do seem to “move to the middle” upon having

secured their parties’ nomination for the election. Wish to

steal away “swing voters” who’d vote for competitor o.w.

• Think of Pepsi and Coke.

Whether spectrum is color, fizziness, sweetness, size, they

have chosen the same location – virtually identical sodas.

1-9© Sergio Turner

Location - limitations

• What if there are more than two competitors?

• What if there are multiple dimensions, not just one spectrum?

E.g. cereals. Makers do not all market “middle cereal,” there is great variety in

many dimensions: price, vitamins, sweetness, crunchiness, colors, size, …

• What if a player can wait to observe the other’s location?• What if a player can wait to observe the other’s location?

• Important not to extrapolate conclusions here to settings that

are not alike

1-10© Sergio Turner

Partnership

• Players: Two partners working on a joint project

• Strategies: Each player chooses a level of effort, Si = [0,4].

• Payoffs: ui = profit/2 – cost of effort

profit := 4(x+y+cxy) where x,y are efforts & 0<c< ¼

cost of effort := square of effortcost of effort := square of effort

u1(x,y) = 2(x+y+c∙xy) – x2

u2(x,y) = 2(x+y+c∙xy) – y2

Each player raises profit by raising own effort

Complementarity: one’s effort made more effective by other’s


Partnership – best responses

• For a belief of player 1 about 2’s effort yε[0,4], let y* = expecn

• 1’s payoff 2(x+y+cxy) – x2 has expectation 2(x+y*+cxy*) – 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):

• Note, BR function of belief only through the expecn y*.

See that if 1 expects 2 to exert higher effort, 1’s b.r. is higher

• By symmetry,

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* *

1(y ) 1BR x cy= = +

* *

2 ( ) 1BR x y cx= = +

© Sergio Turner

Partnership – best responses

• BRi(e) = 1 +ce

Since eε[0,4], BRi(e)ε[1,1+4c]

So Bi = [1,1+4c]

• Player j knows i’s strats off• Player j knows i’s strats off

[1,1+4c] are dominated

• So reduced strategy space is

R1i = [1,1+4c] (recall c< ¼ )


Partnership – rationalizability

• Applying BRi(e) = 1 +ce in iterations …

… to original [0,4] got R1i = [1,1+4c]

… to R1i , get R2

i = [1+c,1+c+4c2]

… to R2i , get R2

i = [1+c+c2,1+c+c2+4c3]

• Since |c|<1, these series converge to 1/(1-c) & 4c^(k+1)->0

• Conclude: (x,y)=(1/(1-c), 1/(1-c)) sole rationalizable profile

• Note: This also intersection of BR’s: 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+

= =

= + ∑ ∑

© Sergio Turner

Partnership - discussion

• Rationalizability concept predicts both exert effort 1/(1-c)

• 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

−+ + = + =

− − −

− −

• Is this efficient? No. Maximizing joint payoff profit –x2 – y2:

• Efficiency requires more effort than rationalizable. Individuals

fail to internalize that own effort makes other’s more effective

• 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

− −= − =

− − −

2 1

1 2 1x y

c c= = >

− −

© 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.


Social unrest

• Players: Uniformly distributed in interval [0,1]

Near 0 are “apathetic,” near 1 are “zealots”

• Strategies: For each player iε[0,1], Si ={stay home H, protest P}

• Payoffs: Let x = fraction of people who choose to protest

ui(H,x) = 4x-2

ui(P,x) = 2(4x-2) + αi

• α>0 parameterizes how intensely protesting raises payoff

Note, zealots enjoy protesting more. Protesting worse if low x


Social unrest – best response

• Zealotry induces protesting, threshold falls with belief x & α

• 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 =−≥⇔≥⇔ α/)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 α>2.

• Case α=1. If player i believes most stay home (x< ¼), then

T>1>=i, so i‘s b.r. is H. Conversely, if x>½, then T<0<=i, so i‘s

b.r. is P. So for all players both H,P are b.r., i.e. Bi = {H,P}, and

any profile is rationalizable.

• Low α (protesting not so exciting) & strategic uncertainty

cause no protest, even if all better off protesting. 1-18© Sergio Turner

Social unrest – rationalizability

• Case α=3.

• Consider i> 2/3. Then whatever x, T<=2/3 & i’s b.r. is P: Bi={P}

Consider i<2/3. Depending on x, b.r. is either, so Bi={P,H}

• Delete H from all i>2/3. KNR implies all believe x>1/3.

THRESHOLDxixHuxPu ii =−≥⇔≥⇔ α/)42(),(),(b.r a is P


But then T ≤2/9, so P becomes dominant for players i>2/9


But then T = -10/27, so b.r. is P for all players i

• Sole rationalizable profile is “all protest”

• Conclude: High α makes social unrest sole rationalizable profl


Social unrest - discussion

• If α is low (<2 actually), then any profile is rationalizable,

including for all to stay home.

Protesting’s benefit, if any, comes from belief x and not own

type, and combo of low beliefs and inaction is “rational”

• If α is high (>2 actually), then “all protest” is sole • If α is high (>2 actually), then “all protest” is sole

rationalizable profile: social unrest guaranteed.

• Factors that raise α: Culture sympathetic to protesters, news

coverage of protesters fair/positive, police respectful, mayor

announces support, ….

Protest organizers often do work on these dimensions.


CHAPTER 8 Location, Partnership & Social Unrest

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