23506 Chap.2 Best Response, Strict vs Weak Dominance, and Iterated Elimination
Best Response, Strict vs Weak Dominance, and Iterated Elimination
A best response is any feasible action that maximises a player’s payoff against fixed opposing strategies. It can be unique or set-valued when several actions tie at the maximum. In a matrix, scan Row’s payoff down each column and Column’s payoff across each row. This keeps the opponent fixed and preserves all ties. Best-response maps later support pure Nash checking, mixed-strategy supports and continuation equilibria inside dynamic games.
Dominance makes a comparison across every possible opposing choice. Strategy a strictly dominates b when a gives strictly more everywhere. Weak dominance allows equal payoff in some cases, requires no loss anywhere and at least one strict gain. Strictly dominated actions can be removed safely in the standard finite analysis. Weak deletion is more delicate because ties can support equilibria and the final survivor can depend on deletion order. A deletion ledger should name the player, removed strategy, dominating strategy or mixture and the currently surviving opponent actions.
Iterated elimination repeats dominance checks in the reduced game. Redraw the active rows and columns after each round; a strategy may become dominated only after an opposing action disappears. Stop when no permitted dominance relation remains and switch to best-response or mixed-equilibrium analysis instead of forcing further deletion. The second-price auction application shows weak dominance through deviation cases: overbidding can force a loss-making win, while underbidding can forfeit a profitable win; truthful bidding is weakly dominant in the stated private-value benchmark.
What this chapter covers
- 01Best-response correspondences and ties
- 02Strict dominance and universal comparison
- 03Weak dominance and equality
- 04Iterated deletion in reduced games
- 05Order sensitivity and mixed dominance
- 06Truthful bidding in a second-price auction
Iterated strict-dominance ledger
- 2Compare Row’s A and B against both columns.
- 2Delete B only after proving strict dominance.
- 2Recompute Column’s comparison using surviving rows A and C.
- 2Record whether another strict deletion exists.
- 2Stop correctly and name the next equilibrium tool.
Key terms
- Best response
- Every payoff-maximising action against fixed opponents.
- Strictly dominant strategy
- A strategy strictly better than every alternative against every opposing profile.
- Strictly dominated strategy
- A strategy strictly worse than another permissible strategy or specified mixture everywhere.
- Weak dominance
- Never worse and sometimes strictly better across all opposing choices.
- IESDS
- Repeated elimination of strictly dominated strategies in the current reduced game.
- Deletion order
- The recorded sequence in which strategies are removed, especially important for weak dominance.
Best Response, Strict vs Weak Dominance, and Iterated Elimination FAQ
Can a player have two best responses?
Yes. Keep both when they tie at the highest payoff.
Does one better cell prove dominance?
No. Dominance requires the comparison for every opposing strategy.
What breaks strict dominance?
Any equality or reversal in the relevant comparison means the proposed pure strategy does not strictly dominate.
What makes weak dominance weak?
The dominating strategy may tie in some cases but must never lose and must win somewhere.
Why can weak deletion be order-sensitive?
Removing tied strategies can remove responses that support another equilibrium or change later comparisons.
Must I continue until one profile remains?
No. Stop when no allowed dominance relationship remains, then use another solution method.
Can a mixture dominate a pure action?
Yes, when the problem permits mixed dominance and the mixture beats the action against every surviving opposing choice.
Is a dominant strategy the same as a best response?
A dominant strategy is a best response regardless of opponents; a best response is defined against one fixed opposing choice.
Why is truthful bidding only weakly dominant?
Some alternative bids produce the same outcome and payoff for particular rival bids, so strict improvement does not hold everywhere.
What should a deletion ledger contain?
Player, deleted strategy, dominating object, payoff comparisons and the current reduced opponent set.
What is a complete dominance proof under exam conditions?
First declare whose payoff coordinate you are reading. Compare the proposed superior strategy with the candidate inferior strategy against every currently feasible opposing action; one favourable column is never enough. Write strict inequality everywhere for strict dominance, or weak inequalities everywhere plus at least one strict inequality for weak dominance. If elimination is iterated, number the rounds and redraw the surviving rows and columns before the next comparison. Name any mixture used to dominate a pure strategy and verify its weighted payoff in every surviving case. Stop when the allowed dominance relation disappears. If several strategies remain, move to mutual best responses or supported mixing rather than treating non-elimination as failure. Finally translate every deletion back to original labels and verify the surviving profile, if unique, is a mutual best response. This format exposes equality, order sensitivity, player ownership and the exact reduced game, so a reader can reproduce every crossed-out row or column.
Exam move
Use a two-colour matrix routine. First mark every best response without deleting anything. Next test strict dominance with one inequality per opposing action. Only then consider weak dominance or a mixed dominating strategy if the question calls for it. For iterated elimination, redraw the reduced matrix after every deletion and number the rounds. Practise one second-price-auction proof by separating value above, below and equal to the highest rival bid. In each drill, finish with a stopping decision: either the survivor is established, or the reduced game needs Nash or mixed analysis. Revise directional vocabulary aloud—dominant is the superior strategy, dominated the inferior one, and best response is conditional on opponents—because a swapped term reverses the claim.
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