Why Coordination Games Can Have More Than One Equilibrium
In some strategic situations, each player does best by choosing the same action as the other player. For example, two firms might benefit from adopting the same technical standard, or two groups might need to select compatible schedules. If either of two matching choices can be sustained, the game may have more than one Nash equilibrium. The challenge is then not just to find equilibria, but to compare what each equilibrium delivers and how vulnerable each is to a unilateral deviation.
This tutorial considers two players, each choosing between actions \(A\) and \(B\). It focuses on strict coordination games: each player prefers to match the other player’s action. The two concepts used to compare their equilibria are payoff dominance and risk dominance. They answer different questions, so they need not identify the same outcome.
Payoff Dominance and Risk Dominance
In a symmetric coordination game, both players receive the same payoff at each matching outcome. Write \(R\) for each player’s payoff at \((A,A)\), \(P\) for each player’s payoff at \((B,B)\), \(S\) for a player’s payoff from choosing \(A\) when the other chooses \(B\), and \(T\) for a player’s payoff from choosing \(B\) when the other chooses \(A\). The payoffs are shown in the matrix below; the first entry in each cell is the row player’s payoff.
| Column: A | Column: B | |
|---|---|---|
| Row: A | \((R,R)\) | \((S,T)\) |
| Row: B | \((T,S)\) | \((P,P)\) |
The strict-coordination conditions are \(R>T\) and \(P>S\). When the column player chooses \(A\), the row player prefers \(A\); when the column player chooses \(B\), the row player prefers \(B\). By symmetry, the same comparisons apply to the column player. Thus both matching outcomes are equilibria.
Payoff dominance compares what players receive at the equilibria. Risk dominance compares how much each player would sacrifice by deviating unilaterally from an equilibrium. At \((A,A)\), a player who switches to \(B\) loses \(R-T\). At \((B,B)\), a player who switches to \(A\) loses \(P-S\). In a symmetric game, comparing these two losses is sufficient; in a game with different payoffs for the two players, the corresponding comparison uses products of the players’ losses.
Why the Risk-Dominance Comparison Works
A useful justification comes from asking how confident a player must be that the other player will choose \(A\). Suppose the player assigns probability \(q\) to the other player choosing \(A\), and \(1-q\) to choosing \(B\). This is a belief about the other player’s action, not an assumption that the other player is randomizing in equilibrium.
The expected payoff from choosing \(A\) is \(qR+(1-q)S\). The expected payoff from choosing \(B\) is \(qT+(1-q)P\). The player prefers \(A\) when the first quantity is larger:
Rearranging gives \(q(R-T)>(1-q)(P-S)\). Since strict coordination gives \(R-T>0\) and \(P-S>0\), the player prefers \(A\) when:
If \(R-T>P-S\), this threshold is less than one-half. The player therefore prefers \(A\) even when their belief that the other player chooses \(A\) is below one-half. That is why \(A\) is called risk-dominant: its equilibrium has the larger deviation-loss margin. Reversing the inequality makes \(B\)’s threshold lower than one-half. This argument explains the comparison; it does not predict with certainty which equilibrium players will choose.
Worked Examples
Worked Example: Payoff-Dominant but Not Risk-Dominant
Two organizations are choosing between compatible standards \(A\) and \(B\). In a hypothetical payoff model, each receives 8 if both adopt \(A\), and each receives 5 if both adopt \(B\). If they choose differently, the organization choosing \(A\) receives 0 and the one choosing \(B\) receives 4. Find the pure-strategy equilibria and compare them.
Method. First check each player’s best response to each action of the other player. Then compare the matching payoffs and the unilateral-deviation losses. This avoids assuming that the equilibrium with the largest payoff must also be the least risky.
The payoff matrix is \((A,A)=(8,8)\), \((A,B)=(0,4)\), \((B,A)=(4,0)\), and \((B,B)=(5,5)\). Against \(A\), a player gets 8 by choosing \(A\) and 4 by choosing \(B\), so matching at \(A\) is a best response. Against \(B\), a player gets 5 by choosing \(B\) and 0 by choosing \(A\), so matching at \(B\) is also a best response. The pure-strategy equilibria are \((A,A)\) and \((B,B)\).
At \((A,A)\), each player gets 8, compared with 5 at \((B,B)\). Therefore \((A,A)\) is payoff-dominant. The loss from a unilateral deviation at \((A,A)\) is \(8-4=4\). The loss from a unilateral deviation at \((B,B)\) is \(5-0=5\). Since \(4<5\), \((B,B)\) is risk-dominant.
Check. The coordination conditions are \(8>4\) and \(5>0\), confirming both strict equilibria. The example shows that payoff dominance and risk dominance can point to different outcomes.
Worked Example: Comparing Risk in an Asymmetric Game
Two businesses must select a compatible booking protocol. A hypothetical payoff matrix is \((A,A)=(10,10)\), \((A,B)=(2,9.5)\), \((B,A)=(9.5,2)\), and \((B,B)=(8,8)\). Identify the equilibria, determine which outcome gives each player more, and compare risk dominance.
Method. For each matching outcome, calculate each player’s loss from changing action alone. When the players’ payoffs are not identical in every cell, compare the products of the two losses at each equilibrium.
At \((A,A)\), if the column player stays with \(A\), the row player gets 10 by choosing \(A\) and 9.5 by choosing \(B\). If the row player stays with \(A\), the column player gets 10 by choosing \(A\) and 9.5 by choosing \(B\). Thus \((A,A)\) is a strict equilibrium. At \((B,B)\), each player gets 8 by staying with \(B\) and 2 by switching to \(A\), so \((B,B)\) is also a strict equilibrium.
Both players prefer \((A,A)\): player 1 receives 10 rather than 8, and player 2 receives 10 rather than 8. Thus \((A,A)\) is payoff-dominant. At \((A,A)\), player 1’s deviation loss is \(10-9.5=0.5\), and player 2’s deviation loss is also \(10-9.5=0.5\). The product is \(0.5(0.5)=0.25\).
At \((B,B)\), player 1’s deviation loss is \(8-2=6\), and player 2’s deviation loss is also \(8-2=6\). The product is \(6(6)=36\). Since \(36>0.25\), \((B,B)\) is risk-dominant. The payoff-dominant equilibrium gives each player more at the outcome, while the risk-dominant equilibrium has much larger losses from unilateral deviations.
Check. The best-response comparisons at both matching outcomes are strict, and the risk comparison uses the product of each player’s loss, not the sum of their payoffs.
Worked Example: Finding the Payoff Threshold for Risk Dominance
A symmetric game has payoffs \((A,A)=(x,x)\), \((A,B)=(1,5)\), \((B,A)=(5,1)\), and \((B,B)=(6,6)\). Find the condition on \(x\) for \((A,A)\) to be an equilibrium, the condition for it to be payoff-dominant, and the condition for it to be risk-dominant. State what happens at \(x=12\).
Method. Apply the equilibrium, payoff, and deviation-loss comparisons separately. Keeping these conditions distinct prevents the common error of treating “higher equilibrium payoff” as equivalent to “risk-dominant.”
For \((A,A)\) to be a strict equilibrium, a player facing \(A\) must prefer \(A\): \(x>5\). The other equilibrium \((B,B)\) is strict because facing \(B\), a player receives 6 by choosing \(B\) and 1 by choosing \(A\). For \((A,A)\) to be payoff-dominant, its payoff must exceed 6, so \(x>6\).
The loss from deviating at \((A,A)\) is \(x-5\), because a player would receive 5 by switching to \(B\) while the other stays at \(A\). The loss from deviating at \((B,B)\) is \(6-1=5\). Therefore \((A,A)\) is risk-dominant when \(x-5>5\), or \(x>10\). At \(x=10\), neither equilibrium is risk-dominant by this comparison.
For \(x=12\), both matching outcomes are strict equilibria. Since \(12>6\), \((A,A)\) is payoff-dominant. Its deviation loss is \(12-5=7\), compared with 5 at \((B,B)\), so \((A,A)\) is also risk-dominant. The calculations confirm that the conditions are different: payoff dominance begins at \(x>6\), while risk dominance begins at \(x>10\).
Common Mistake: Comparing the Wrong Payoffs
A frequent error is to compare the off-diagonal payoffs directly when deciding risk dominance. In the first worked example, an incorrect calculation might compare the payoff of 0 from choosing \(A\) against \(B\) with the payoff of 4 from choosing \(B\) against \(A\), then conclude that \(A\) is riskier or less attractive. Those are payoffs in mismatch outcomes; they are not, by themselves, the losses from deviating at the two equilibria.
The correct calculation holds the other player’s action fixed at each equilibrium. At \((A,A)\), compare the payoff from staying at \(A\), 8, with the payoff from deviating to \(B\), 4: the loss is \(8-4=4\). At \((B,B)\), compare 5 with 0: the loss is \(5-0=5\). Thus \(B\) is risk-dominant in that example. The equilibrium payoff comparison is separate: \(8>5\), so \(A\) is payoff-dominant.
Interpreting the Comparison
Risk dominance is a comparison of incentives under uncertainty about the other player’s action. It does not guarantee that the risk-dominant equilibrium will occur, and it does not mean that the other equilibrium is not a Nash equilibrium. Likewise, payoff dominance says that one equilibrium gives both players a higher payoff in the symmetric setting; it does not establish how players will coordinate on it.
For a game with different player payoffs, the standard risk comparison is based on products of deviation losses. If the losses at \((A,A)\) are \(L_A^1\) and \(L_A^2\), and those at \((B,B)\) are \(L_B^1\) and \(L_B^2\), compare \(L_A^1L_A^2\) with \(L_B^1L_B^2\). The equilibrium with the larger product is risk-dominant under this two-equilibrium comparison. In a symmetric game the two players’ losses are equal, so comparing the products is equivalent to comparing the single-player losses.
Check Your Understanding
For each problem, identify the requested equilibrium comparison and show the relevant payoff or deviation-loss calculation.
- A symmetric coordination game has \((A,A)=(9,9)\), \((A,B)=(1,6)\), \((B,A)=(6,1)\), and \((B,B)=(7,7)\). Name both pure equilibria and determine which is payoff-dominant and which is risk-dominant. (Answer: Both matching outcomes are strict equilibria; \((A,A)\) is payoff-dominant, while \((B,B)\) is risk-dominant because the deviation losses are 3 and 6.)
- A symmetric game has \(R=11\), \(T=4\), \(P=6\), and \(S=2\). Verify the coordination conditions and identify the payoff-dominant and risk-dominant equilibria. (Answer: Both matching outcomes are strict equilibria; \((A,A)\) is both payoff-dominant and risk-dominant, since \(11>6\) and \(11-4=7>6-2=4\).)
- A two-player game has \((A,A)=(12,9)\), \((A,B)=(3,8)\), \((B,A)=(10,2)\), and \((B,B)=(7,6)\). Verify both matching equilibria, identify the payoff-dominant outcome, and compare the deviation-loss products. (Answer: Both are strict equilibria; \((A,A)\) gives both players more. Its loss product is \((12-10)(9-8)=2\), while \((B,B)\)'s is \((7-3)(6-2)=16\), so \((B,B)\) is risk-dominant.)
- A symmetric game has \((A,A)=(x,x)\), \((A,B)=(2,5)\), \((B,A)=(5,2)\), and \((B,B)=(7,7)\). Find the conditions for \((A,A)\) to be a strict equilibrium, payoff-dominant, and risk-dominant. (Answer: Strict equilibrium if \(x>5\); payoff-dominant if \(x>7\); risk-dominant if \(x>10\).)
- In a strict symmetric coordination game, the losses from deviating at \((A,A)\) and \((B,B)\) are 4 and 9, respectively. Which equilibrium is risk-dominant, and what does that conclusion not guarantee? (Answer: \((B,B)\) is risk-dominant because \(9>4\); this does not guarantee that players will choose \((B,B)\), nor does it remove \((A,A)\) as an equilibrium.)