The Encyclopedia of Economic Principals

Chapter 9

Strategic Foundations and Equilibrium Concepts

Predict play by asking what each side would do against the other's best choice.

Four of the chapter's worked examples, made interactive: a randomized fraud monitor, delivery competition that turns from a prisoners dilemma into chicken, coordination without common knowledge, and a correlating signal for freight carriers.

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

Randomize the fraud monitor

How often should the retailer watch online orders when the fraudster picks the unwatched channel?

The retailer's guarantee is the smaller of the two lines. The best guarantee sits where they cross, which leaves the fraudster indifferent between channels.

Equation, written in LaTeX: 4p=2(1-p),

Equation, written in LaTeX: 4q=2(1-q),

Scroll sideways for the whole equation

The retailer monitors online with probability p, in store otherwise. Catching an online attack avoids a loss (4, in $ thousand, in the book); catching an in-store attack avoids 2. v is the guaranteed expected loss avoided.

Predict first. If catching online fraud is worth more, should the retailer monitor online more or less often?

Your prediction

Choose an example

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Figure: Randomize the fraud monitor. Lines 4p and 2(1 - p) against p, crossing at p = 1/3 with value 4/3.
Loss avoided by catching online fraud ($ thousand): 4, Fraudster sees the assignment: No
Constructed example: the chapter's hypothetical retailer (payoffs 4 and 2, hidden or observed); online values of 2 and 6 are added for comparison.

Calculated values

Retailer's p*
1/3 (0.333)
Fraudster's q*
1/3 (0.333)
Value v
4/3 ($1,333)
Guaranteed with no mixing
0

Equalizing 4p = 2(1 - p) gives (4 + 2)p = 2, so p* = 2 / 6 = 1/3 and v = 4 x 1/3 = 4/3, about $1,333 of avoided loss. The fraudster's q solves 4q = 2(1 - q), so q* = 1/3. Any pure choice guarantees 0.

Worked steps

  1. 4p = 2(1 - p)
  2. p* = 2 / (4 + 2) = 1/3
  3. v = 4 x 1/3 = 4/3 = 1.333
  4. q* = 1/3

Use the idea

Randomize inspections when the target can respond to a predictable pattern, and keep the draw hidden.

Where the conclusion applies

A zero-sum game, one analyst, simultaneous hidden choices and expected-value payoffs.

Check your understanding: With the online catch worth 6, what are p* and v?
6p = 2(1 - p) gives p = 1/4; v = 6 x 1/4 = 1.5, or $1,500.

Chapter 9 source: section "Minimax theorem".

Demonstration 2 of 4

From a prisoners dilemma to a game of chicken

How does the payoff when both chains offer delivery change the equilibria?

While the shared payoff is positive, offering beats not offering whatever the rival does: a prisoners dilemma. Once it is negative, each wants the opposite of the rival, which gives two pure equilibria and a mixed one.

Equation, written in LaTeX: p(-1)+(1-p)6=6-7p,

Equation, written in LaTeX: p(0)+(1-p)4=4-4p.

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NorthMart (rows) and SouthFoods (columns) choose Offer same-day delivery or Not, with profits in millions. One offering alone earns 6 and the other 0; neither offering gives 4 each. p is the rival's probability of offering.

Predict first. At a payoff of 1 each when both offer, is Offer still dominant?

Your prediction

Choose an example

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Figure: From a prisoners dilemma to a game of chicken. Payoff matrix with (O, O) paying 2 each; equilibrium cells shaded: (O, O).
Payoff each when both offer: 2
Constructed example: the chapter's hypothetical grocery chains (payoffs 2 and -1 at (O, O)); payoffs 1 and -3 are added for comparison.

Calculated values

Pure equilibria
(O, O)
Offer dominant
yes
Mixed equilibrium
none
Payoff each at (N, N)
4

Against O, Offer pays 2 > 0; against N it pays 6 > 4. Offer is dominant, so (O, O) is the only equilibrium, with 2 each against 4 at (N, N): each chain gives up 4 - 2 = 2. There is no mixed equilibrium because Offer strictly dominates.

Worked steps

  1. Best reply to O: Offer (2 vs 0)
  2. Best reply to N: Offer (6 vs 4)
  3. Pure equilibria: (O, O)

Use the idea

Ask whether a rival's move makes yours more or less attractive before predicting a single outcome.

Where the conclusion applies

Simultaneous one-shot choices, known payoffs and risk-neutral firms.

Check your understanding: At a payoff of -3 each when both offer, what are the mixed offer probability and payoff?
6 - 9p = 4 - 4p gives p = 2/5; payoff 4 - 4 x 0.4 = 2.4.

Chapter 9 source: section "Nash equilibrium and existence".

Demonstration 3 of 4

When private email is not enough

Should a firm adopt a new standard when it is not sure the other firm heard?

Without a public signal each firm weighs the chance the other did not hear. A large loss from adopting alone makes even a small doubt decisive.

Equation, written in LaTeX: 0.9(5)+0.1(-50)=-0.5.

Equation, written in LaTeX: 0.9(5)+0.1(-10)=3.5>0.

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Both adopting earns 5 each; a lone adopter loses the solo loss; waiting pays 0. d is the probability the other firm's email arrived, so that it adopts.

Predict first. Does 99% delivery reliability rescue adoption with the $50 loss?

Your prediction

Choose an example

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Figure: When private email is not enough. Expected payoff of adopting against delivery probability; at d = 0.9 it is -0.50, break-even 0.909.
Email delivery probability: 0.9, Loss from adopting alone: 50
Constructed example: the chapter's hypothetical standard (delivery 0.9, losses 50 and 10); deliveries of 0.8 and 0.99 are added for comparison.

Calculated values

Expected payoff of adopting
-0.50
Break-even delivery probability
0.909
Decision
wait

Adopting pays 0.9 x 5 + 0.10 x (-50) = 4.50 - 5.00 = -0.50, against 0 for waiting, so Alpha should wait. Adoption needs d above 50 / (5 + 50) = 0.909.

Worked steps

  1. 0.9 x 5 = 4.50
  2. 0.10 x 50 = 5.00
  3. Expected = 4.50 - 5.00 = -0.50
  4. Break-even d = 50 / 55 = 0.909

Use the idea

Make coordination announcements public when failing to coordinate is costly.

Where the conclusion applies

The other firm adopts exactly when informed, risk neutrality and independent deliveries.

Check your understanding: With delivery 0.8 and loss 10, should Alpha adopt?
0.8 x 5 + 0.2 x (-10) = 4 - 2 = 2 > 0, so adopt.

Chapter 9 source: section "Common knowledge".

Demonstration 4 of 4

A traffic light for freight carriers

Will carriers follow a port's random schedule of peak and off-peak slots?

A recommendation is information about what the other was told. Off-peak is obeyed only if the other is likely enough to be on Peak that switching risks a collision.

Equation, written in LaTeX: \frac{0.4}{0.4+0.2}=\frac23.

Equation, written in LaTeX: \frac23(2)+\frac13(4)=\frac83,

Equation, written in LaTeX: \frac23(0)+\frac13(6)=2.

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Each carrier chooses Peak (P) or Off-peak (O). Payoffs: 4 each at (O, O), 6 and 2 at (P, O), and the collision payoff at (P, P). The port recommends (P, O) and (O, P) with probability x each and (O, O) otherwise.

Predict first. Does putting more weight on (O, O) make obedience after an Off-peak recommendation easier or harder?

Your prediction

Choose an example

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Figure: A traffic light for freight carriers. Bars of obey and deviate payoffs; after Off-peak obey 8/3 against deviate 2.
Probability of each (P, O) and (O, P): 0.4, Collision payoff (P, P): 0
Constructed example: the chapter's hypothetical port (x = 0.4, collision payoffs 0 and 3); x of 0.3 and 0.45 is added for comparison.

Calculated values

P(other told Peak | told Off-peak)
2/3 (0.667)
Obey after Off-peak
8/3 (2.667)
Deviate after Off-peak
2 (2.000)
Schedule obeyed
yes
Expected payoff each
4.00

After Off-peak the other was told Peak with probability 0.4 / (0.4 + 0.2) = 2/3. Obeying pays 2/3 x 2 + 1/3 x 4 = 8/3; switching to Peak pays 2/3 x 0 + 1/3 x 6 = 2. Obedience pays at least as much, and after a Peak recommendation 6 beats 4, so the schedule is obeyed. Expected payoff is 0.4 x 6 + 0.4 x 2 + 0.2 x 4 = 4.00.

Worked steps

  1. pi = 0.4 / (0.4 + 0.2) = 2/3
  2. Obey = 2/3 x 2 + 1/3 x 4 = 8/3
  3. Deviate = 2/3 x 0 + 1/3 x 6 = 2
  4. Told Peak: obey 6 against deviate 4
  5. Obeyed: yes

Use the idea

A public signal can coordinate rivals on outcomes better than independent mixing, if each recommendation is in the recipient's interest.

Where the conclusion applies

Both carriers trust the device, know the schedule and maximize expected payoff.

Check your understanding: With x = 0.3 and a collision payoff of 0, is the schedule obeyed?
pi = 0.3 / 0.7 = 3/7; obey = 3/7 x 2 + 4/7 x 4 = 22/7 = 3.14; deviate = 4/7 x 6 = 24/7 = 3.43; it fails.

Chapter 9 source: section "Correlated equilibrium".