1
Coordination games
Each player does best by matching the other, and there is more than one match to choose. The payoffs reward getting on the same page and do not, in the plainest cases, say which page.
Lesson · Games and the Social Contract · One-shot games
Coordination, Conventions and the Stag Hunt
In a coordination game both players want to match, and there is more than one way to match. A convention is an equilibrium that is kept going by each player's expectation that the others keep it, with no promise needed. In the stag hunt the best match is also the riskier one, and the lab gives the belief at which the risk is worth taking.
coordination: match, and several matches pay
convention: stable because others conform
two equilibria: one pays most, one is safest
stag is best only above a belief threshold
Which equilibrium, and why
Core ideas
What this lesson establishes
A game with two equilibria raises a question the table cannot answer: which one? Three ideas give the question some structure.
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Convention
A convention is a regularity that is an equilibrium of a coordination game that recurs, and that is followed because the others follow it. Another regularity would have done as well.
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Payoff dominance against risk dominance
One equilibrium may pay both players more while another costs them less if the other player turns out to be elsewhere. The two properties can pick different equilibria.
The material
Three coordination games
Driving, the stag hunt, and a game where the players disagree about which match is best.
The driving game is the plainest coordination problem. Two drivers meet on a road and each chooses to keep to the left or to the right. If they choose the same side they pass safely and each gets 1. If they choose differently each gets 0. The lab marks two equilibria, left with left and right with right, and gives them identical payoffs.
Definition · Convention
A convention is a way of behaving, in a coordination game that keeps coming up, such that everyone follows it, everyone expects the others to follow it, and everyone prefers to follow it provided the others do. Some other way of behaving would have served as well.
Driving on the right is a convention in this sense in every country that does it, and so is driving on the left. Nothing in the table prefers either. What keeps a country on one side is each driver's expectation of the others, and that expectation is in turn kept going by the fact that each does best conforming. It does not need anyone to have promised anything, which is why the term “agreement” is the wrong model: an agreement could start a convention, and a custom with no origin in any agreement can be one.
The stag hunt has the same two matched cells with payoffs that differ. Two hunters can each go for a stag or for a hare. A stag takes both of them to bring down and is worth 4 to each; a hare can be taken alone and is worth 3. A hunter who goes for a stag alone gets nothing.
column Stag column Hare
row Stag 4, 4 0, 3
row Hare 3, 0 3, 3
Stag with stag and Hare with Hare are both equilibria. The first pays both players more, so it is payoff-dominant; the second is Pareto-dominated by the first. But Hare guarantees 3 whatever the other does, and Stag risks 0. One standard measure of that risk compares what each player would lose by deviating: from Stag, Stag a deviator loses 1, and from Hare, Hare a deviator loses 3. The equilibrium whose product of losses is larger, here 3·3 = 9 against 1·1 = 1, is risk-dominant. That is Hare, Hare.
The belief at which stag is worth it
Let q be the probability the hunter gives to the other hunting stag. Going for the stag pays 4q, since it succeeds only when the other joins. Going for the hare pays 3 whatever happens. So stag is the better choice exactly when 4q ≥ 3, that is when q ≥ 3/4.
Three-quarters is the number the lab prints as the mixed equilibrium, now read as a threshold on belief. A hunter who is only fifty-fifty about the partner should go for the hare, and a pair who both give stag a probability of nine in ten can sustain it. The table does not say which belief the pair will have. That depends on what each has seen the other do, which is the part of the problem a convention answers and a table cannot.
A third game, the battle of the sexes, adds disagreement about which match is better. One player prefers the opera and the other the football; both prefer going together to going apart. With the opera worth 3 to one and 2 to the other, and the football the reverse, the lab marks both matches as equilibria and prints a mix in which the row player chooses the opera with probability 3/5 and the column player with probability 2/5. At that mix each expects only 6/5, less than the 2 either gets at the match they like less. The move from a coordination problem to a bargaining problem is already visible: even a convention now favours someone.
Interactive
A game in a table
best responses, equilibria, dominance and efficiency, computed from the payoffs
Method
Reading a coordination game
Five questions about a table with two matching cells.
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List the pure equilibria
Mark the best responses. A coordination game has at least two doubly marked cells.
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Compare their payoffs
If one pays everyone more, it is payoff-dominant. If the payoffs are the same, the table is silent.
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Find what each player risks
Ask what a player gets in each equilibrium if the other player plays the other strategy.
4
Compute the belief threshold
With belief q that the other plays the good strategy, write both payoffs and solve for the q that makes them equal.
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Say what would settle the choice
Precedent, a signal, or a history of play. These are things outside the table, and a convention is what they leave behind.
Matching, risk and belief
Score 0 / 0Choose the answer you can justify.
When is the stag worth hunting
Payoffs: stag with stag 4, a lone stag hunter 0, hare 3 whatever happens.
q = chance the other hunts stag
Hunt stag: 4·q + 0·(1 - q) = 4q
Hunt hare: 3·q + 3·(1 - q) = 3
Stag is at least as good when 4q >= 3
so q >= 3/4
The lab's mixed equilibrium, 3/4 for each player, is the same number: it is the belief that leaves the other player indifferent between stag and hare. Read as a threshold on belief it says how much trust the better equilibrium needs.
Common mistakes
1
Thinking a convention is an agreementIn the driving game the two sides pay the same, and no one signs anything for either. What makes each side an equilibrium is that conforming is a best response to the others conforming. An agreement can start such a regularity, but a convention that began in custom is as much a convention as one that began in a treaty.
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Taking the payoff-dominant equilibrium to be the one rational players chooseStag with stag pays 4 each, against 3 for hare with hare, and rational hunters still end up at hare when each doubts the other. Rationality alone does not pick between equilibria; expectation does.
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Hunting stag on an even chanceA fifty-fifty belief gives stag an expectation of 2 against the hare's certain 3. The better equilibrium needs a belief of three in four, which is the reason trust is hard to start and easy to lose.
Check your understanding
✓
Finish when you can compute the belief at which the risky equilibrium is worth it.
Given a two-by-two coordination game, name its pure equilibria, say which is payoff-dominant and which is risk-dominant, and compute the probability of the partner's cooperation above which the better equilibrium is the better bet.
Note: Risk dominance has several definitions. The one here, the larger product of deviation losses, is the standard one for two-by-two games and agrees with the belief threshold: the risk-dominant equilibrium is the one that is best for the wider range of beliefs.