Chapter 12: Coordination, Conflict, Evolution, and Strategic Networks. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Assurance in the stag hunt

How confident must a producer be that its partner will join before choosing the shared network?

Both all-Network and all-private are equilibria. Which one a producer picks depends on its belief, and a guarantee for a lone adopter lowers the belief needed.

Equation, written in LaTeX: 12p-2(1-p)=14p-2,

$Equation, written in LaTeX: 14p-2\geq7 \Longleftrightarrow p\geq\frac{9}{14}\approx0.643.$

$Equation, written in LaTeX: p^*=\frac{7}{(12-7)+7}=\frac{7}{12}\approx0.583.$

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Mutual Network pays 12 each and the private system 7 whatever the partner does (thousands). A lone Network adopter gets -2, or 0 with the cooperative's guarantee. p is the belief the partner joins.

Predict first. Does the guarantee change the choice at a belief of 0.55?

Your prediction No, the private system still wins Yes, Network now wins

Lone adopter payoff

Belief partner joins

Figure: Assurance in the stag hunt. Network payoff line against belief, threshold 0.643; at p = 0.7 it pays 7.80 against 7. — Lone adopter payoff: -2 (no guarantee), Belief partner joins: 0.7
Constructed example: the chapter's hypothetical producers (lone payoff -2 or 0, beliefs 0.55, 0.60 and 0.70); all values are from the book.

Network payoff: 7.80
Private payoff: 7
Threshold belief: 0.643
Choice: Network

Without a guarantee, Network pays 12 x 0.7 + (-2) x 0.30 = 8.40 + (-0.60) = 7.80 against 7, so the producer chooses: Network. Network pays off once p reaches (7 - (-2)) / (12 - (-2)) = 0.643.

Worked steps

12 x 0.7 = 8.40
-2 x 0.30 = -0.60
Network = 7.80 against 7
Threshold = 9 / 14 = 0.643

Use the idea

To launch a shared system, insure early adopters against being alone rather than only promising high joint returns.

Where the conclusion applies

Two producers, a known payoff table and risk neutrality.

Check your understanding: With the guarantee and p = 0.55, is Network chosen?

12 x 0.55 = 6.6 < 7, so no; the threshold is 7/12 = 0.583.

Chapter 12 source: section "Stag-hunt game".

Demonstration 2 of 4

Hawks and doves settle at a mix

What share of aggressive players can a population sustain?

When Hawks are common, they mostly meet each other and pay the fight cost, so Doves do better; when Hawks are rare, they mostly meet Doves and do better. The mix where the two earn the same is stable.

Equation, written in LaTeX: U_H=x(-20)+(1-x)60=60-80x,

Equation, written in LaTeX: U_D=(1-x)30=30-30x.

Equation, written in LaTeX: U_H-U_D=30-50x.

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V is the value contested and C the cost of a fight. Two Hawks each get (V - C) / 2; Hawk takes V from a Dove; two Doves share V. x is the Hawk share of the population.

Predict first. As fighting gets costlier, does the stable Hawk share rise or fall?

Your prediction Fall Rise

Contest value V

Conflict cost C

Figure: Hawks and doves settle at a mix. Hawk and Dove payoff lines against the Hawk share; stable share 0.60 with V = 60, C = 100. — Contest value V: 60, Conflict cost C: 100
Constructed example: the chapter's hypothetical population (V = 60, C = 100 and 50); V = 40 and C = 80 and 150 are added for comparison.

Hawk vs Hawk payoff: -20.0
Stable Hawk share x*: 0.60
Payoff at x*: 12.00
Type: mixed

Hawk meeting Hawk gets (60 - 100) / 2 = -20.0. U_H = x(-20.0) + (1 - x)60 and U_D = (1 - x)30 are equal at x* = 60 / 100 = 0.60, where both earn 12.00. At 0.55 Hawk earns 16.00 against 13.50, so Hawks spread; at 0.65 Hawk earns 8.00 against 10.50, so they shrink.

Worked steps

Hawk vs Hawk = (60 - 100) / 2 = -20.0
x* = 60 / 100 = 0.60
Payoff at x* = (1 - 0.60) x 30 = 12.00

Use the idea

Expect aggression to persist at a level that falls as conflict grows more costly.

Where the conclusion applies

Random pairing, a large population and payoffs that drive reproduction or imitation.

Check your understanding: With V = 60 and C = 150, what is the stable Hawk share?

x* = V / C = 0.40 (U_H = 60 - 105x and U_D = 30 - 30x are equal at 0.4).

Chapter 12 source: section "Evolutionarily stable strategy".

Demonstration 3 of 4

A new road that slows everyone

Can adding a free shortcut make every commuter slower?

The shortcut tempts every driver onto both congestible edges. Each driver's choice is best given the others, but together they load the slow edges and everyone loses.

$Equation, written in LaTeX: \frac{2{,}000}{100}+45=65\text{ minutes}.$

$Equation, written in LaTeX: \frac{4{,}000}{100}+\frac{4{,}000}{100}=80.$

Equation, written in LaTeX: 4{,}000(80)=320{,}000,

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Edges S to A and B to T take x / 100 minutes with x cars; edges A to T and S to B take 45 minutes. The connector A to B takes 0 minutes. Each commuter picks the fastest route.

Predict first. At 3,000 commuters, does opening the link help, hurt, or do nothing?

Your prediction Nothing: 60 minutes either way Helps Hurts

Commuters

Connector A to B

Minutes per commuter: 80
Total commuter-minutes: 320,000
Minutes with no connector: 65
Change from opening the link: +60,000

With the connector everyone takes S, A, B, T: 4,000 / 100 + 4,000 / 100 = 80 minutes, while a lone switch to an old route would take 85. Total 4,000 x 80 = 320,000 against 260,000 without it, so the link hurts everyone.

Worked steps

No connector: 2,000 / 100 + 45 = 65 minutes
Total with no connector: 4,000 x 65 = 260,000
Connector: 40 + 40 = 80 minutes
Lone switch: 40 + 45 = 85 minutes
Change: 320,000 - 260,000 = 60,000

Use the idea

Before adding capacity to a network, check how route choices will shift in equilibrium.

Where the conclusion applies

Identical commuters, fixed demand and selfish routing to a Wardrop equilibrium.

Check your understanding: With 3,000 commuters, what are travel times with the connector closed and open?

Closed: 15 + 45 = 60. Open: all use the shortcut at 30 + 30 = 60, old routes would take 30 + 45 = 75; no change.

Chapter 12 source: section "Braess paradox".

Demonstration 4 of 4

How bad is selfish routing?

How much longer do commuters travel when each picks a road for themselves?

Each commuter ignores the delay they add to others on road A, so too many use it. The planner moves some to B even though B is slower for them.

Equation, written in LaTeX: C(x)=x^2+(1-x).

Equation, written in LaTeX: C'(x)=2x-1,

$Equation, written in LaTeX: \operatorname{PoA}=\frac{1}{3/4}=\frac43\approx1.333.$

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A unit mass of commuters splits between road A, with latency equal to its share x, and road B, with constant latency b. Social cost is average travel time.

Predict first. With road B at 1.5, is the price of anarchy above or below 4/3?

Your prediction Below Above

Road B latency

Equilibrium cost: 1.0000
Optimal share on A: 0.50
Optimal cost: 0.7500
Price of anarchy: 1.333

In equilibrium all traffic uses A, whose latency 1 is at most 1, so cost is 1. The planner sets 2x - 1 = 0, so x = 0.50, with cost 0.50^2 + 1 x 0.50 = 0.2500 + 0.5000 = 0.7500. PoA = 1.0000 / 0.7500 = 1.333.

Worked steps

Equilibrium cost = 1.0000
Optimum x = min(1 / 2, 1) = 0.50
Optimal cost = 0.2500 + 0.5000 = 0.7500
PoA = 1.0000 / 0.7500 = 1.333

Use the idea

Compare equilibrium and optimal costs to judge how much a congestion charge could gain.

Where the conclusion applies

Nonatomic commuters, fixed demand and linear latency on road A.

Check your understanding: With road B at 0.5, what are the equilibrium cost, the optimal cost and the PoA?

Equilibrium puts 0.5 on A, cost 0.5; the planner puts 0.25 on A, cost 0.0625 + 0.375 = 0.4375; PoA = 0.5 / 0.4375 = 1.143.

Chapter 12 source: section "Price of anarchy".

Assurance in the stag hunt

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Hawks and doves settle at a mix

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

A new road that slows everyone

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

How bad is selfish routing?

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies