23506 Chap.3 Nash Equilibrium in Matrix Games: The Check Method, Focal Points, Payoff vs Risk Dominance
Nash Equilibrium in Matrix Games: The Check Method, Focal Points, Payoff vs Risk Dominance
A pure-strategy Nash equilibrium is a profile in which every player’s chosen action is a best response to the others. The check method is deliberately mechanical: for each column, mark every row payoff at its maximum; for each row, mark every column payoff at its maximum. Cells carrying both players’ marks are pure Nash equilibria. Ties must be retained, and every unmarked cell fails because at least one player can improve unilaterally.
Equilibrium is a stability concept, not a welfare or fairness test. A stable profile can be inefficient, unequal or one of several possibilities. Coordination games often contain multiple pure equilibria, anti-coordination games often contain asymmetric equilibria, and cyclic games may have no pure equilibrium. These patterns help recognise a search problem but never replace payoff checks. A 3×3 game uses the same scan, with every maximum retained before any support calculation.
Selection concepts operate after the equilibrium set is known. A focal point can coordinate expectations through salience, convention or precedent without changing payoffs. Payoff dominance compares equilibria and selects one that gives all players more when such a comparison exists. Risk dominance concerns robustness when players are uncertain about coordination and should be established through a belief threshold or deviation-loss calculation. Self-fulfilling beliefs connect an expectation to an action that produces an outcome consistent with the original expectation.
What this chapter covers
- 01Pure Nash equilibrium as mutual best response
- 02Two-colour matrix check method
- 03Coordination, anti-coordination and cycles
- 04Focal points and selection evidence
- 05Payoff dominance versus stability
- 06Risk dominance and self-fulfilling beliefs
Complete equilibrium scan
- 2Mark Row’s best response to L and R.
- 2Mark Column’s best response to U and D.
- 2Intersect the response marks.
- 2Test all remaining cells for a profitable deviation.
- 2Compare the equilibrium payoffs without confusing selection with existence.
Key terms
- Nash equilibrium
- A profile of mutually best-responding strategies.
- Check method
- A systematic scan marking each player’s entire best-response set.
- Focal point
- A salient coordination candidate that can organise expectations among equilibria.
- Payoff dominance
- An equilibrium comparison in which one profile gives all players strictly higher payoffs.
- Risk dominance
- Robustness of a coordination equilibrium to uncertainty about others’ choices.
- Self-fulfilling belief
- An expectation that induces actions producing an outcome consistent with that expectation.
Nash Equilibrium in Matrix Games: The Check Method, Focal Points, Payoff vs Risk Dominance FAQ
Is every highest-total-payoff cell Nash?
No. Each player must resist a unilateral deviation; total payoff is not the criterion.
Can a Nash equilibrium involve ties?
Yes. Chosen actions need only be among the best responses, not uniquely best.
Does no pure equilibrium mean no equilibrium?
Not in finite games. A mixed-strategy equilibrium still exists, though it may require later methods.
What does find all require?
Retain all tied best responses, scan every cell and later test relevant mixed supports.
Can a focal point create equilibrium?
No. It can select among profiles already supported by incentives.
Is payoff dominance always available?
No. Equilibria may distribute gains differently so neither benefits every player.
How is risk dominance established?
Derive the belief probability at which a player is indifferent, or use the specified deviation-loss criterion.
Can an equilibrium be inefficient?
Yes. Nash rules out profitable unilateral deviation, not Pareto improvement by coordinated change.
How do I scan a 3×3 table?
Mark all Row maxima within each column and all Column maxima within each row, then find intersections.
Why use two colours?
It prevents payoff-coordinate confusion and makes mutual marks and ties visible.
How do I distinguish equilibrium existence from equilibrium selection?
First ignore salience and scan incentives. For every opponent action, mark the entire best-response set, retaining tied maxima. Intersect the players’ response marks and list every pure Nash profile. If a finite game lacks a pure intersection, note that mixed analysis remains; do not call the game equilibrium-free. Only after the equilibrium set is complete should selection begin. A focal point uses labels, convention or precedent to coordinate expectations but does not alter payoffs. Payoff dominance ranks equilibria only when one gives every player more. Risk dominance asks which equilibrium is more robust to uncertainty and should be supported by the specified belief threshold or loss comparison. Self-fulfilling beliefs require a loop from expectation to optimal action to an outcome validating that expectation. Report each conclusion on its own line. This ordering prevents a salient or high-payoff profile from being misreported as the only Nash equilibrium and makes explicit which result follows from incentives versus an added coordination assumption.
Exam move
Practise the check method on matrices whose equilibria are not visually diagonal. Always list Row responses by column, Column responses by row and then the intersection. Add one sentence excluding every remaining profile through at least one profitable deviation. Once the set is secure, apply a separate selection layer: identify a possible focal cue, compare equilibrium payoffs and derive any risk threshold requested. Build counterexamples to the claims that equilibrium is unique, efficient or fair. For 3×3 games, use a fixed scan order so ties are never skipped. A strong closed-book rehearsal reports response sets, equilibrium profiles and selection interpretation in that order.
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