23506 Chap.4 Nash Equilibrium Applications: Cournot, Bertrand, Entry and the Demand Game
Nash Equilibrium Applications: Cournot, Bertrand, Entry and the Demand Game
Applied Nash problems often hide a familiar mutual-best-response check inside a richer strategy space. In the discrete Cournot model, firms simultaneously choose quantities from 0,1,2,3. Inverse demand is p=10−xA−xB and profit is (10−xA−xB)xi−4.5xi. For every rival quantity, calculate the firm’s payoff across feasible own quantities and retain the argmax. The response schedules intersect at (2,2), where price is 6 and each firm earns 3.
The demand game asks players to claim shares of a resource. If total compatible claims are below 100, a player can raise its claim; equilibrium compatible claims therefore exhaust 100. The boundary can contain infinitely many equilibria, including unequal allocations. A changed excess-demand default changes deviation guarantees and must be analysed afresh. In the 100-firm entry application, revenue 100/x and cost 6 require adjacent integer checks: 16 active firms are profitable, while a seventeenth would not be.
Bertrand price competition changes the strategic variable from quantity to price. Undercutting logic depends on homogeneous products, marginal cost, demand allocation under ties, capacity and feasible prices. The comparison with Cournot or Stackelberg is meaningful only when other primitives are held fixed. Guess-and-verify is legitimate when the guess is followed by every player’s deviation check; model names or symmetry alone do not establish equilibrium.
What this chapter covers
- 01Discrete Cournot profit and reaction schedules
- 02Cournot equilibrium (2,2) and profit verification
- 03Demand-game resource boundary
- 04Integer entry and adjacent-firm checks
- 05Bertrand prices and undercutting
- 06Controlled comparison of quantity, price and timing
Rebuild the Cournot equilibrium
- 2Fix B at 2 and calculate A’s profit at every feasible quantity.
- 2Identify A’s best response, retaining any tie.
- 2Repeat for B with A fixed at 2.
- 2Calculate equilibrium price, revenue and cost.
- 2State the profile, outcome and no-deviation conclusion.
Key terms
- Cournot competition
- Firms choose quantities simultaneously and price follows total output.
- Reaction schedule
- A player’s best response for each opposing action.
- Strategic substitutes
- Actions for which a higher opposing choice tends to lower one’s best response.
- Bertrand competition
- Firms compete through prices under stated demand and cost assumptions.
- Entry equilibrium
- A market count where active firms stay and an additional firm does not profitably enter.
- Resource boundary
- Profiles that exhaust a fixed amount without activating an excess-demand penalty.
Nash Equilibrium Applications: Cournot, Bertrand, Entry and the Demand Game FAQ
Why is (2,2) Cournot Nash?
Each firm earns more at quantity 2 than at 0,1 or 3 when the rival supplies 2.
What are price and profits at (2,2)?
Price is 6 and each firm earns profit 3 under the stated inverse demand and cost.
Why not use a continuous first-order condition?
The course example restricts quantities to four integers, so equilibrium follows direct feasible-payoff comparison.
Must demand-game equilibrium be equal?
No. Compatible claims must exhaust 100, but many unequal boundary profiles can be equilibria under the basic zero rule.
Why can someone receive zero in demand equilibrium?
If others already claim 100, raising a zero claim triggers excess demand and still returns zero, so there may be no profitable deviation.
Why is the entry count 16?
100/16 exceeds cost 6, while revenue with 17 active firms falls below 6. Both stay and entry conditions are checked.
What is the strategic variable in Bertrand?
Price, whereas Cournot uses quantity.
Does Bertrand always imply marginal-cost pricing?
Only under its maintained assumptions; differentiation, capacity, price grids or tie rules can change the result.
What makes guess-and-verify valid?
The guessed profile is tested against every feasible unilateral deviation for each player.
How should models be compared?
Hold demand, cost and feasible choices fixed where possible, then identify the changed strategic variable or timing and compare outcomes.
How do I verify an applied Nash result rather than memorise it?
Reconstruct the payoff rule from primitives. For Cournot, hold rival quantity fixed, calculate total output, price, revenue, cost and own profit for every feasible quantity, then repeat for the rival. The equilibrium is an intersection of response schedules; recompute price and both profits at the intersection. For entry, distinguish profit with the current number of active firms from the payoff of one additional entrant, using adjacent integer counts. For demand, prove that slack invites a higher claim and test how the exact excess-demand rule blocks or rewards deviation. For Bertrand, state the product, marginal cost, feasible price set, capacity and demand allocation under a tie before using undercutting. If comparing models, hold demand and cost fixed and identify whether quantity, price or timing changed. Finish with an economic sentence naming the force that removes profitable deviation. A symmetric or familiar-looking outcome remains only a candidate until every player’s complete feasible deviation set has been checked.
Exam move
Reconstruct the Cournot table from its profit function rather than memorising (2,2). Use a four-column ledger containing own quantity, total output, price and own profit, then repeat for every rival quantity to see the reaction schedule. For entry, calculate incumbent and marginal-entrant profit at adjacent integer counts. For demand, prove both directions of the resource-boundary logic: slack invites an increase, while excess-demand consequences can block one. For Bertrand, state product, cost, tie-demand and capacity assumptions before applying undercutting. End each problem with a sentence naming the unilateral force that stabilises the result.
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