Assurance Games and Cooperation Risk

Why Assurance Games Depend on Confidence

Some economic decisions pay off well only when enough other decision-makers make the same choice. A firm may invest in equipment for a new industry standard, for example, but earn a high return only if a rival invests too. If both firms hold back, each may avoid a loss, even though both could do better by investing together. The central strategic issue is not simply whether cooperation is beneficial; it is whether a player expects the other to cooperate.

An assurance game is a coordination game in which each player prefers to cooperate when the other cooperates, but prefers to avoid cooperation when the other does not. This creates two stable pure-strategy outcomes: mutual cooperation and mutual noncooperation. The first can yield the higher payoff, while the second can remain attractive when a player is uncertain about the other’s choice.

Definition: In a symmetric two-player assurance game, each player chooses Cooperate (\(C\)) or Defect (\(D\)). Write each payoff pair in row-player, column-player order. The outcomes are \((C,C)=(R,R)\), \((C,D)=(S,T)\), \((D,C)=(T,S)\), and \((D,D)=(P,P)\). The assurance conditions are \(R>T\) and \(P>S\): cooperating is better against a cooperator, while defecting is better against a defector. The usual payoff-dominance condition \(R>P\) says mutual cooperation pays each player more than mutual defection. If mutual cooperation is also to maximize total payoff relative to a one-sided outcome, require \(2R>T+S\).

The two best-response conditions explain the name. A player who is confident the other will cooperate prefers \(C\); a player who expects the other to defect prefers \(D\). Neither action is strictly dominant under the assurance conditions. This differs from a prisoner’s dilemma, where defection is a strictly dominant action regardless of the other player’s choice.

Beliefs and the Cooperation Threshold

Let \(q\) be the probability a player assigns to the other player choosing \(C\). For this calculation, assume the player is risk-neutral over the stated payoffs and evaluates actions by their expected payoff. The expected payoff from choosing \(C\) is \(qR+(1-q)S\); from choosing \(D\), it is \(qT+(1-q)P\). Cooperate when the first is at least as large as the second.

qR+(1-q)S \geq qT+(1-q)P

Rearranging gives the threshold. The coefficient on \(q\) is positive because \(R>T\), and \(P-S\) is positive because \(P>S\). Thus a player cooperates when the belief that the other will cooperate is at or above \(q^*\), defects below it, and is indifferent exactly at it.

q(R-T) \geq (1-q)(P-S) \quad\Longrightarrow\quad q \geq q^*=\frac{P-S}{(R-T)+(P-S)}

This threshold also identifies the symmetric mixed-strategy equilibrium: if each player independently chooses \(C\) with probability \(q^*\), the other player is indifferent between \(C\) and \(D\). The pure equilibria follow from the same best-response conditions. At \((C,C)\), neither player gains by switching because \(R>T\); at \((D,D)\), neither gains by switching because \(P>S\). The two off-diagonal outcomes are not equilibria: the player choosing \(C\) against \(D\) would gain by switching to \(D\), since \(P>S\).

Theorem: Under \(R>T\) and \(P>S\), an assurance game has two pure-strategy Nash equilibria, \((C,C)\) and \((D,D)\), and a symmetric mixed-strategy equilibrium in which each player chooses \(C\) with probability \(q^*\). The belief threshold \(q^*\) is the probability of the other player choosing \(C\) that makes a player indifferent; it is not itself a guarantee that the other player will cooperate.

Worked Examples

Worked Example: A Shared Industry Standard

Two firms are deciding whether to adopt a compatible technology standard. Payoffs are hypothetical profits in millions of dollars. If both adopt, each earns 16; if neither adopts, each earns 5. If one adopts alone, the adopter earns 1 and the non-adopter earns 7. Find the pure equilibria, the cooperation threshold, and which pure equilibrium is risk-dominant.

Method. Assign \(R=16\), \(P=5\), \(S=1\), and \(T=7\). Check the assurance conditions: \(R>T\) because \(16>7\), and \(P>S\) because \(5>1\). Also \(R>P\), so each firm earns more at mutual adoption than at mutual non-adoption.

At \((C,C)\), a unilateral switch changes the deviator’s payoff from 16 to 7, so the switch is unprofitable. At \((D,D)\), a unilateral switch changes the deviator’s payoff from 5 to 1, also unprofitable. The two pure equilibria are therefore \((C,C)\) and \((D,D)\).

The cooperation threshold is \(q^*=(P-S)/[(R-T)+(P-S)]=(5-1)/[(16-7)+(5-1)]=4/(9+4)=4/13\), or approximately \(0.3077\). A firm should adopt if it assigns a probability of at least about 30.77% to the other firm adopting.

Check the threshold directly at \(q=0.25\). The expected payoff from adopting is \(0.25(16)+0.75(1)=4+0.75=4.75\). The expected payoff from not adopting is \(0.25(7)+0.75(5)=1.75+3.75=5.50\), so non-adoption is better. At \(q=0.40\), adoption gives \(0.40(16)+0.60(1)=6.40+0.60=7.00\); non-adoption gives \(0.40(7)+0.60(5)=2.80+3.00=5.80\). Adoption is better, as the threshold predicts.

For risk dominance, compare the losses from a unilateral deviation at each equilibrium. At \((C,C)\), the deviator loses \(R-T=16-7=9\). At \((D,D)\), the deviator loses \(P-S=5-1=4\). The loss from disrupting mutual cooperation is larger, so \((C,C)\) is risk-dominant in this symmetric game. The calculation avoids comparing payoffs across outcomes where the other player’s action also changes.

Worked Example: Separating Assurance From a Prisoner’s Dilemma

Two suppliers are considering whether to keep a costly capacity reserve available. Suppose the hypothetical payoffs, in thousands of dollars, are \(R=11\), \(T=14\), \(P=6\), and \(S=1\). Determine whether this is an assurance game or a prisoner’s dilemma, and find the equilibrium.

Method. Compare each player’s payoffs for the same action by the opponent. If the opponent cooperates, cooperating pays 11 while defecting pays 14. If the opponent defects, cooperating pays 1 while defecting pays 6. Defection pays more in both cases: \(T>R\) and \(P>S\). Thus \(D\) is strictly dominant, which is the defining incentive pattern of a prisoner’s dilemma, not an assurance game.

The expected payoff from cooperation when the other player cooperates with probability \(q\) is \(11q+(1-q)1=1+10q\). The expected payoff from defection is \(14q+(1-q)6=6+8q\). Their difference, cooperation minus defection, is \((1+10q)-(6+8q)=-5+2q\). Since \(0\leq q\leq1\), this difference is at most \(-5+2=-3<0\). Cooperation is never a best response, regardless of the belief \(q\).

Each player therefore chooses \(D\), and the unique Nash equilibrium is \((D,D)\), with payoff 6 to each. Mutual cooperation would pay 11 to each, but neither player has an individual incentive to maintain cooperation when defection gives a higher payoff against either action. This is the essential contrast: in an assurance game, confidence can change the best response; in this prisoner’s dilemma, it cannot.

Worked Example: Cooperation Pays More, but Defection Is Safer

Two regional carriers are deciding whether to support a shared booking platform. Their hypothetical payoffs, in millions of dollars, are \(R=13\), \(T=10\), \(P=8\), and \(S=2\). Find the equilibria and threshold, then compare payoff dominance with risk dominance.

Method. Since \(13>10\) and \(8>2\), this is an assurance game. Since \(R>P\), mutual cooperation pays 13 each, more than the 8 each at mutual defection. The pure equilibria are \((C,C)\) and \((D,D)\): at the first, switching lowers a player’s payoff from 13 to 10; at the second, switching lowers it from 8 to 2.

The belief threshold is \(q^*=(8-2)/[(13-10)+(8-2)]=6/(3+6)=6/9=2/3\), approximately \(0.6667\). A player needs to believe the other will cooperate with probability at least two-thirds before cooperation is a best response.

At \(q=0.50\), the expected payoff from cooperation is \(0.50(13)+0.50(2)=6.50+1=7.50\). Defection yields \(0.50(10)+0.50(8)=5+4=9\), so defection is better. At \(q=0.70\), cooperation yields \(0.70(13)+0.30(2)=9.10+0.60=9.70\), while defection yields \(0.70(10)+0.30(8)=7+2.40=9.40\), so cooperation is better.

The deviation loss at \((C,C)\) is \(R-T=13-10=3\). The loss at \((D,D)\) from switching to cooperation is \(P-S=8-2=6\). Since 6 exceeds 3, \((D,D)\) is risk-dominant, even though \((C,C)\) is payoff-dominant. These terms answer different questions: payoff dominance compares the payoffs at the equilibria, while risk dominance compares how costly it is to depart unilaterally from each equilibrium.

Why Risk Dominance Uses These Deviation Losses

In a symmetric assurance game, the risk-dominance comparison can be justified by asking how much a player loses by making the wrong unilateral choice at each equilibrium. At \((C,C)\), the other player remains at \(C\); the deviator’s payoff therefore changes from \(R\) to \(T\), giving a loss of \(R-T\). At \((D,D)\), the other player remains at \(D\); switching from \(D\) to \(C\) changes the deviator’s payoff from \(P\) to \(S\), giving a loss of \(P-S\).

In a symmetric two-player game, each player faces the same loss at either equilibrium. The standard risk-dominance comparison uses the product of the players’ deviation losses. Those products are \((R-T)^2\) for \((C,C)\) and \((P-S)^2\) for \((D,D)\). Because both losses are positive, comparing their squares is equivalent to comparing the losses themselves. Thus \((C,C)\) is risk-dominant if \(R-T>P-S\); \((D,D)\) is risk-dominant if \(R-T<P-S\); and neither is risk-dominant by this comparison if the losses are equal.

The threshold gives the same result. Since \(q^*=(P-S)/[(R-T)+(P-S)]\), it is below one-half exactly when \(R-T>P-S\). In that case, cooperation is a best response even when the player assigns less than a 50% probability to the other cooperating. This link is useful, but the belief threshold and risk dominance are not interchangeable concepts: the threshold describes a player’s best response to a belief, while risk dominance compares the equilibria.

Common Mistakes and Economic Significance

A frequent error is to calculate deviation losses by comparing payoffs in cells that change both players’ actions. That does not describe a unilateral deviation. The other player’s action must remain fixed.

Common mistake: For the platform example with \(R=13\), \(T=10\), \(P=8\), and \(S=2\), an incorrect comparison might use \(R-S=13-2=11\) as the loss at \((C,C)\), and \(T-P=10-8=2\) as the loss at \((D,D)\). It would incorrectly label cooperation risk-dominant. The correction holds the other player’s action fixed: from \((C,C)\), the payoff falls from \(R\) to \(T\), a loss of \(13-10=3\); from \((D,D)\), it falls from \(P\) to \(S\), a loss of \(8-2=6\). Defection is risk-dominant in this example.

Another trap is to assume that an outcome with the higher payoff must be the equilibrium. A payoff comparison does not establish a best response: the player must compare actions while holding the other player’s choice fixed. Assurance games matter in economic settings because expectations, communication, and credible coordination can affect which equilibrium is reached. The payoff matrix alone establishes possible equilibria, but it does not tell us which one players will select.

When analyzing an unfamiliar matrix, use a consistent sequence: identify who receives each payoff; check the two assurance inequalities; find the pure equilibria; derive the belief threshold if uncertainty is relevant; and only then compare payoff dominance and risk dominance. If \(R>T\) fails or \(P>S\) fails, do not apply the assurance-game conclusions mechanically.

Check Your Understanding

For each problem, show the payoff comparisons before identifying equilibria or thresholds.

A pair of wholesalers has hypothetical payoffs \(R=20\), \(T=12\), \(S=3\), and \(P=7\). Identify the pure equilibria, calculate the cooperation threshold, and state which pure equilibrium is risk-dominant. Is cooperation a best response when the probability of the other wholesaler cooperating is \(0.40\)? (Answer: The assurance conditions hold; pure equilibria are \((C,C)\) and \((D,D)\); \(q^*=4/(8+4)=1/3\); \((C,C)\) is risk-dominant because \(R-T=8>P-S=4\); at \(q=0.40\), cooperate.)
Two distributors have hypothetical payoffs \(R=15\), \(T=11\), \(S=4\), and \(P=7\). Find the threshold and risk-dominant equilibrium. Which action is a best response at \(q=0.40\), and which at \(q=0.50\)? (Answer: Both pure equilibria exist; \(q^*=3/(4+3)=3/7\approx0.4286\); \((C,C)\) is risk-dominant because \(4>3\); defect at \(q=0.40\), cooperate at \(q=0.50\).)
A pair of retailers faces hypothetical payoffs \(R=12\), \(T=9\), \(S=1\), and \(P=6\). Find the threshold, risk-dominant equilibrium, and best response when \(q=0.60\). (Answer: Both \((C,C)\) and \((D,D)\) are equilibria; \(q^*=5/(3+5)=5/8=0.625\); \((D,D)\) is risk-dominant because \(P-S=5>R-T=3\); at \(q=0.60\), defect.)
In a hypothetical two-firm game, \(R=10\), \(T=13\), \(S=0\), and \(P=4\). Decide whether the game is an assurance game or a prisoner’s dilemma. Use expected payoffs to verify whether cooperation can be a best response for any \(q\in[0,1]\), and state the equilibrium. (Answer: It is a prisoner’s dilemma because defection strictly dominates; cooperation’s expected payoff is \(10q\), defection’s is \(4+9q\), and their difference is \(q-4<0\) for all \(q\in[0,1]\); the unique equilibrium is \((D,D)\).)
A pair of contractors has hypothetical payoffs \(R=18\), \(T=12\), \(S=0\), and \(P=4\). Calculate the threshold, identify the risk-dominant equilibrium, and state the best response when \(q=0.35\). (Answer: Both pure equilibria exist; \(q^*=4/[(18-12)+(4-0)]=4/10=0.40\); \((C,C)\) is risk-dominant because \(R-T=6>P-S=4\); at \(q=0.35\), defect.)

What You'll Learn