The Encyclopedia of Economic Principals

Chapter 10

Dynamic and Sequential Games

Solve from the end, and test every threat at the moment it would be carried out.

Four of the chapter's worked examples, made interactive: a credible or empty fight threat, the patience needed to hold a high price, entry against an incumbent of unknown strength, and the cost of a standards war.

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

Is the fight threat credible?

Should an entrant believe an incumbent's threat to fight?

Solve from the last move back. A threat is credible only if carrying it out is the mover's best choice when the time comes.

Equation, written in LaTeX: 2>-1,

Equation, written in LaTeX: 3>0.

Equation, written in LaTeX: -1>-3,

Equation, written in LaTeX: 0>-2.

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Payoffs are (entrant, incumbent). Stay out gives (0, 5). After entry, Fight gives (-2, incumbent's fight payoff) and Accommodate gives (3, incumbent's accommodate payoff).

Predict first. If Accommodate pays the incumbent 0 and Fight pays -1, does entry happen?

Your prediction

Choose an example

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Figure: Is the fight threat credible? Entry game tree; the incumbent's best reply is Accommodate and the entrant's choice is Enter, giving (3, 2).
Incumbent payoff from Accommodate: 2, Incumbent payoff from Fight: -1
Constructed example: the chapter's hypothetical entry game (Accommodate 2 or -3, Fight -1); Accommodate 0 and Fight 1 are added for comparison.

Calculated values

Incumbent after entry
Accommodate
Entrant
Enter
Payoffs (entrant, incumbent)
(3, 2)
Fight threat credible
no

After entry the incumbent compares Accommodate 2 with Fight -1: 2 > -1, so it chooses Accommodate. The entrant compares 3 from Enter with 0 from Stay out and chooses Enter. The outcome pays (3, 2); the incumbent's gain from its choice after entry is 2 - (-1) = 3.

Worked steps

  1. Incumbent: Accommodate 2 vs Fight -1: Accommodate
  2. Entrant: Enter 3 vs Stay out 0: Enter
  3. Outcome (3, 2)

Use the idea

Judge a rival's threat by what it would earn from carrying it out after you have acted.

Where the conclusion applies

Perfect information, known payoffs and players who maximize their own payoff at every node.

Check your understanding: With Accommodate at 0 and Fight at 1, what is the outcome?
At the last node 1 > 0, so the incumbent fights; the entrant compares -2 with 0 and stays out; payoffs (0, 5).

Chapter 10 source: section "Backward induction".

Demonstration 2 of 4

How patient must firms be to hold the high price?

When does no single deviation from grim-trigger pricing pay?

Checking each state for a single profitable deviation is enough. Patience raises the value of future cooperation that a deviation throws away.

Equation, written in LaTeX: V_C=\frac{8}{1-0.7}=26.667.

Equation, written in LaTeX: V_D=12+0.7\frac{5}{1-0.7}=23.667.

Equation, written in LaTeX: \frac{8}{1-\delta}\geq12+\frac{5\delta}{1-\delta},

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Mutual High pays 8 each period, a lone Low pays the deviation payoff (12 in the book), mutual Low pays 5. delta is the discount factor. After any Low both play Low forever.

Predict first. Does raising the deviation payoff from 12 to 14 make delta 0.6 insufficient?

Your prediction

Choose an example

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Figure: How patient must firms be to hold the high price? Cooperation and deviation values against delta crossing at 0.571; at 0.7 they are 26.667 and 23.667.
Discount factor delta: 0.7, One-period deviation payoff: 12
Constructed example: the chapter's hypothetical pricing game (delta 0.7 and 0.5, deviation 12); delta 0.6 and 0.9 and deviation payoffs 10 and 14 are added for comparison.

Calculated values

V_C
26.667
V_D
23.667
Margin V_C - V_D
3.000
Threshold delta
0.571
Cooperation sustained
yes

V_C = 8 / (1 - 0.7) = 26.667. V_D = 12 + 0.7 x 5 / (1 - 0.7) = 12 + 11.667 = 23.667. The margin is 3.000, so cooperation holds. Cooperation needs delta of at least (12 - 8) / (12 - 5) = 0.571. In the punishment state Low earns 5 and High earns 2 against Low with the same future, so no one-shot deviation pays there.

Worked steps

  1. V_C = 8 / 0.3 = 26.667
  2. V_D = 12 + 11.667 = 23.667
  3. Margin = 26.667 - 23.667 = 3.000
  4. Threshold = 4 / 7 = 0.571

Use the idea

Test a pricing or cooperation agreement by asking whether one defection, followed by the stated punishment, would pay.

Where the conclusion applies

Infinite repetition, perfect monitoring and grim-trigger punishment.

Check your understanding: With a deviation payoff of 14, what is the threshold, and does delta 0.6 sustain cooperation?
8 >= 14(1 - delta) + 5 delta gives delta >= 6/9 = 0.667; at 0.6, V_C = 20 < V_D = 14 + 7.5 = 21.5, so no.

Chapter 10 source: section "One-shot deviation principle".

Demonstration 3 of 4

Entry against an incumbent of unknown strength

How does the entrant's belief about the incumbent's type decide entry?

Each type's action after entry must be optimal for that type. The prior only weights those actions; it cannot make a Weak incumbent's fight credible.

Equation, written in LaTeX: 0.30(-1)+0.70(3)=-0.30+2.10=1.80>0.

Equation, written in LaTeX: 0.80(-1)+0.20(3)=-0.80+0.60=-0.20<0.

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q is the prior probability the incumbent is Strong. The entrant earns -1 if fought and 3 if accommodated; Out pays 0. Strong earns 2 from fighting and 1 from accommodating; Weak earns -2 and 1.

Predict first. At q = 0.75, what is the entrant's expected payoff from entering?

Your prediction

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Figure: Entry against an incumbent of unknown strength. Entry payoff line 3 - 4q crossing zero at 0.75; at q = 0.3 it is 1.80.
Prior probability incumbent is Strong: 0.3
Constructed example: the chapter's hypothetical incumbent (priors 0.30 and 0.80); priors 0.50 and 0.75 are added for comparison.

Calculated values

Expected payoff of Enter
1.80
Entrant's choice
Enter
Strong incumbent
fights (2 > 1)
Weak incumbent
accommodates (1 > -2)

Strong fights because 2 > 1 and Weak accommodates because 1 > -2, whatever the prior. Entry pays 0.3 x (-1) + 0.70 x 3 = -0.30 + 2.10 = 1.80 against 0 for Out, so the entrant's choice is: Enter.

Worked steps

  1. 0.3 x (-1) = -0.30
  2. 0.70 x 3 = 2.10
  3. E[Enter] = -0.30 + 2.10 = 1.80
  4. Choice: Enter

Use the idea

Estimate how likely a rival is to be the type that would actually fight, and enter when the weighted payoff is positive.

Where the conclusion applies

Two types, a known prior, and sequential rationality at every information set.

Check your understanding: With q = 0.50, should the entrant enter?
0.5 x (-1) + 0.5 x 3 = 1.0 > 0, so enter.

Chapter 10 source: section "Sequential equilibrium".

Demonstration 4 of 4

The cost of a standards war

How long does a war of attrition last, and how much does it burn?

In the symmetric equilibrium each vendor concedes at the rate that makes the other indifferent, hV = c. Expected spending then uses up the whole prize.

Equation, written in LaTeX: h=\frac{c}{V}=\frac{3}{120}=0.025 \text{per week}.

Equation, written in LaTeX: \frac{1}{2h}=\frac{1}{0.05}=20\text{ weeks}.

Equation, written in LaTeX: 1-e^{-0.025}\approx0.02469,

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V is the value of becoming the standard and c each vendor's weekly cost, in thousands. h is the constant concession hazard of each vendor.

Predict first. If both the prize and the weekly cost double, does the expected duration change?

Your prediction

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Figure: The cost of a standards war. Survival curve of the contest with hazard 0.0500; expected duration 20 weeks.
Prize value V ($ thousand): 120, Weekly cost per vendor ($ thousand): 3
Constructed example: the chapter's hypothetical vendors (V = 120, c = 3, 10-week deadline); prizes of 60 and 240 and a weekly cost of 6 are added for comparison.

Calculated values

Hazard h
0.0250
Mean concession time
40 weeks
Expected duration
20 weeks
One-week clock probability
2.47%
Expected combined cost
120
Cost bound with deadline
60

h = 3 / 120 = 0.0250 per week, so each vendor's mean concession time is 1 / h = 40 weeks and the contest lasts 1 / (2h) = 20 weeks on average. One vendor's clock rings within a week with probability 1 - e^(-0.0250) = 0.02469. Expected combined cost is 2 x 3 x 20 = 120, the whole prize. A 10-week deadline caps it at 2 x 3 x 10 = 60.

Worked steps

  1. h = 3 / 120 = 0.0250
  2. Duration = 1 / (2 x 0.0250) = 20 weeks
  3. One-week probability = 1 - e^(-0.0250) = 0.02469
  4. Cost = 2 x 3 x 20 = 120
  5. Deadline bound = 2 x 3 x 10 = 60

Use the idea

Expect a contest for a valuable prize to absorb resources of about the prize's value unless something, such as a deadline, ends it.

Where the conclusion applies

Symmetric vendors, a constant weekly cost, no deadline in the base case, and memoryless concession clocks.

Check your understanding: With V = 240 and c = 3, what are h, the expected duration and the expected combined cost?
h = 3 / 240 = 0.0125; duration 1 / (2h) = 40 weeks; cost 2 x 3 x 40 = 240, again equal to V.

Chapter 10 source: section "War-of-attrition game".