Rabin (1993): Intention-Based Reciprocity
• The model incorporates three stylized facts:
(A) People are willing to sacrifice their own material well-being to help those
who are being kind (positive reciprocity).
(B) People are willing to sacrifice their own material well-being to punish
those who are being unkind (negative reciprocity).
(C) Both motivations (A) and (B) have a greater effect on behavior as the
material cost of sacrificing becomes smaller (fairness motives are “a
normal good”).
• Note that this differs from the inequity aversion model, where utility depends
only on payoff differences and not on other individuals' actions or intentions.
• Evidence from, e.g., the “mini ultimatum game” suggests that payoffs and
intentions matter.
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Battle-of-the-Sexes Game
• Consider the following version of a “battle-of-the-sexes game,” with 𝑋𝑋 > 0:
Player 2
Player 1
Opera
Boxing
Opera
2X, X
0, 0
Boxing
0, 0
X, 2X
• The battle-of-the-sexes game captures a situation in which two players prefer
to play either (opera, opera) or (boxing, boxing) rather than not coordinating.
Still, player 1 prefers (opera, opera), and player 2 prefers (boxing, boxing).
• The players must decide simultaneously without being able to communicate.
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Nash equilibria in pure strategies
Player 2
Player 1
Opera
Boxing
Opera
2X, X
0, 0
Boxing
0, 0
X, 2X
• Assume first that the players’ utility depends only on their actions (standard
game theory) and that they care only about their own material payoffs (narrow
self-interest)
 two Nash-equilibria in pure strategies: (opera, opera) and (boxing, boxing)
• Assume now that utility can depend on players’ actions and directly on beliefs
(psychological game theory) and that players have social preferences
 additional “fairness equilibria” can exist
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Example of a Fairness Equilibrium (1)
• Assume that player 1 (“he”) cares not only about his own payoff but,
depending on player 2’s (“she”) motives, also about player 2's payoff.
 If player 2 intentionally helps player 1, then player 1 wants to help player 2
 if player 2 intentionally hurts player 1, then player 1 wants to hurt player 2
• Suppose player 1 believes that
1) player 2 is playing boxing, and
2) player 2 believes that player 1 is playing boxing ( second-order belief)
• Then player 1 concludes that player 2 is choosing an action that helps both
players (player 2 playing opera would hurt both players, i.e., also player 2).
• Player 1 thus perceives player 2 as neither generous nor mean.
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Example of a Fairness Equilibrium (2)
• As player 2 is not being either generous (kind) or mean (unkind) to player 1,
neither stylized fact (A) nor (B) applies.
• Thus, player 1 will be neutral about his effect on player 2 and will pursue his
material self-interest, which means playing boxing (given his beliefs).
• The same argument can be repeated for player 2, so that (boxing, boxing) is an
equilibrium: if it is common knowledge that (boxing, boxing) will be the
outcome, then each player is maximizing their utility by playing “boxing.”
 (boxing, boxing) is a fairness equilibrium of the game
• Recall that (boxing, boxing) is also a conventional Nash equilibrium of the game.
• Due to the symmetry of the game, the same reasoning applies regarding the
strategy combination (opera, opera), which is both a “fairness equilibrium” and a
Nash equilibrium.
239
Example of a Fairness Equilibrium (3)
• Now, to see the importance of fairness, suppose player 1 believes that
1) player 2 is playing boxing, and
2) player 2 believes that player 1 is playing opera (second-order belief)
• Now, player 1 concludes that player 2 is lowering her own payoff to hurt him and,
therefore, feels hostility toward player 2 and wants to harm her.
• Player 1 might then be willing to sacrifice his material payoff and play opera rather
than boxing.
• If both players have a strong enough emotional reaction to each other's behavior,
then (opera, boxing) is a fairness equilibrium. (We will derive this formally.)
• Note that (opera, boxing) is not a Nash equilibrium of the game.
• The players’ expectations are crucial in the fairness equilibrium: player 1’s utility
depends not only on his actions but also directly on his beliefs about player 2’s
intentions.
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Model of Intention-Based Reciprocity
• Consider a two-player, normal form game with (mixed) strategy sets 𝑆𝑆1 and 𝑆𝑆2
for players 1 and 2, derived from finite pure strategy sets 𝐴𝐴1 and 𝐴𝐴2 .
• Let 𝜋𝜋𝑖𝑖 : 𝑆𝑆1 × 𝑆𝑆2 → ℝ be player 𝑖𝑖’s material payoffs.
• A “psychological game” is constructed from the above “material game” by
assuming that each player’s subjective expected utility when she chooses her
strategy will depend on three factors:
1. her strategy
2. her beliefs about the other player's strategy choice (belief)
3. her beliefs about the other player's beliefs about her strategy (2nd order
belief)
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Notation
• Strategies chosen by the two players:
 strategy of player 1: 𝑎𝑎1 ∈ 𝑆𝑆1
 strategy of player 2: 𝑎𝑎2 ∈ 𝑆𝑆2
• (1st order) beliefs:
 player 2’ beliefs about what strategy player 1 is choosing: 𝑏𝑏1 ∈ 𝑆𝑆1
 player 1’ beliefs about what strategy player 2 is choosing: 𝑏𝑏2 ∈ 𝑆𝑆2
• 2nd order beliefs:
 player 1’s beliefs about what player 2 believes player 1’s strategy is: 𝑐𝑐1 ∈ 𝑆𝑆1
 player 2’s beliefs about what player 1 believes player 2’s strategy is: 𝑐𝑐2 ∈ 𝑆𝑆2
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Kindness function (1)
• The “kindness function” 𝑓𝑓𝑖𝑖 (𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 ) measures how kind player 𝑖𝑖 is being to player 𝑗𝑗.
• If player 𝑖𝑖 believes that player 𝑗𝑗 is choosing strategy 𝑏𝑏𝑗𝑗 , how kind is player 𝑖𝑖 being
by choosing 𝑎𝑎𝑖𝑖 ?
• By choosing 𝑎𝑎𝑖𝑖 , player 𝑖𝑖 is choosing the payoff pair 𝜋𝜋𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 , 𝜋𝜋𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑎𝑎𝑖𝑖 from the
set of all payoffs feasible, given player 𝑗𝑗 is choosing strategy 𝑏𝑏𝑗𝑗 (which player 1
believes she does).
• Set of all feasible payoffs: Π 𝑏𝑏𝑗𝑗 ≡
𝜋𝜋𝑖𝑖 𝑎𝑎, 𝑏𝑏𝑗𝑗 , 𝜋𝜋𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑎𝑎
• Let 𝜋𝜋𝑗𝑗ℎ 𝑏𝑏𝑗𝑗 be player 𝑗𝑗’s highest payoff in Π 𝑏𝑏𝑗𝑗
𝑎𝑎 ∈ 𝑆𝑆𝑖𝑖
• Let 𝜋𝜋𝑗𝑗ℓ 𝑏𝑏𝑗𝑗 be player 𝑗𝑗’s lowest payoff in Π 𝑏𝑏𝑗𝑗 among the points that are Paretoefficient in 𝛱𝛱 𝑏𝑏𝑗𝑗
243
Kindness function (2)
• Let 𝜋𝜋𝑗𝑗𝑒𝑒 𝑏𝑏𝑗𝑗 =
𝜋𝜋𝑗𝑗ℎ 𝑏𝑏𝑗𝑗 + 𝜋𝜋𝑗𝑗ℓ 𝑏𝑏𝑗𝑗
2
be the “equitable payoff”
 Provides a “reference point” against which it is measured how generous
player 𝑖𝑖 is being to player 𝑗𝑗.
• Let 𝜋𝜋𝑗𝑗𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑗𝑗 be the worst possible payoff for player 𝑗𝑗 in Π 𝑏𝑏𝑗𝑗
• Definition 1: Player 𝑖𝑖’s kindness (by choosing 𝑎𝑎𝑖𝑖 ) to player 𝑗𝑗 is given by:
𝑓𝑓𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 ≡
𝜋𝜋𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑎𝑎𝑖𝑖 − 𝜋𝜋𝑗𝑗𝑒𝑒 𝑏𝑏𝑗𝑗
𝜋𝜋𝑗𝑗ℎ 𝑏𝑏𝑗𝑗 − 𝜋𝜋𝑗𝑗𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑗𝑗
If 𝜋𝜋𝑗𝑗ℎ 𝑏𝑏𝑗𝑗 − 𝜋𝜋𝑗𝑗𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑗𝑗 = 0, then 𝑓𝑓𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 = 0.
244
Kindness function (3)
• In the following, we disregard the case 𝜋𝜋𝑗𝑗ℎ 𝑏𝑏𝑗𝑗 = 𝜋𝜋𝑗𝑗𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑗𝑗 , where player 𝑗𝑗’s
payoff is independent of player 𝑖𝑖’s action.
• We have 𝑓𝑓𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 = 0 if and only if player 𝑖𝑖, by choosing 𝑎𝑎𝑖𝑖 , gives player 𝑗𝑗 her
equitable payoff.
• If player 𝑖𝑖 gives player 𝑗𝑗 less than her equitable payoff, we have 𝑓𝑓𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 < 0
 This can arise either if player 𝑖𝑖 takes more than his equitable share on the
Pareto frontier or is choosing an inefficient point in Π 𝑏𝑏𝑗𝑗 . In the latter
case, this might be costly for player 1 himself.
• If player 𝑖𝑖 gives player 𝑗𝑗 more than her equitable payoff, we have 𝑓𝑓𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 > 0
245
Kindness function (4)
• Let the function 𝑓𝑓�𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 represent player 𝑖𝑖’s beliefs about how kindly player 𝑗𝑗 is
treating him.
• Definition 2: Player 𝑖𝑖’s belief about how kind player 𝑗𝑗 is being to him is given by:
𝑓𝑓�𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 ≡
𝜋𝜋𝑖𝑖 𝑐𝑐𝑖𝑖 , 𝑏𝑏𝑗𝑗 − 𝜋𝜋𝑖𝑖𝑒𝑒 𝑐𝑐𝑖𝑖
𝜋𝜋𝑖𝑖ℎ 𝑐𝑐𝑖𝑖 − 𝜋𝜋𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑖𝑖
If 𝜋𝜋𝑖𝑖ℎ 𝑐𝑐𝑖𝑖 − 𝜋𝜋𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑖𝑖 = 0, then 𝑓𝑓�𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 = 0.
• Due to the normalization of the kindness functions, 𝑓𝑓𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 and 𝑓𝑓�𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 are
bounded and lie in the interval
1
2
1
−1,
2
.
To see this: 𝑓𝑓𝑖𝑖 = if (i) 𝜋𝜋𝑗𝑗ℓ = 𝜋𝜋𝑗𝑗𝑚𝑚𝑚𝑚𝑚𝑚 , in which case 𝜋𝜋𝑗𝑗𝑒𝑒 takes on its lowest possible value, and (ii) 𝜋𝜋𝑗𝑗 = 𝜋𝜋𝑗𝑗ℎ , which
is the kindest move player 𝑖𝑖 can make. Likewise, 𝑓𝑓𝑖𝑖 = −1 if (i) 𝜋𝜋𝑗𝑗ℓ = 𝜋𝜋𝑗𝑗ℎ , in which case 𝜋𝜋𝑗𝑗𝑒𝑒 takes on its highest
246
possible value, and (ii) 𝜋𝜋𝑗𝑗 = 𝜋𝜋𝑗𝑗𝑚𝑚𝑚𝑚𝑚𝑚 , which is the unkindest move player 𝑖𝑖 can make.
Players’ Utility Function
• Each player 𝑖𝑖 chooses 𝑎𝑎𝑖𝑖 to maximize their expected utility
𝑈𝑈𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 = 𝜋𝜋𝑖𝑖 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 + 𝑓𝑓̃𝑗𝑗 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 ⋅ 1 + 𝑓𝑓𝑖𝑖 (𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑗𝑗 )
which incorporates a player’s material payoff and their notion of fairness.
• These preferences capture reciprocity:
 If player 𝑖𝑖 believes that player 𝑗𝑗 is unkind, 𝑓𝑓̃𝑗𝑗 < 0, then player 𝑖𝑖’s wants to
choose an action 𝑎𝑎𝑖𝑖 such that 𝑓𝑓𝑖𝑖 is low or even negative , i.e., player 𝑖𝑖 wants
to be unkind to player 𝑗𝑗.
 Likewise, if 𝑓𝑓̃𝑗𝑗 > 0, then player 𝑖𝑖’s wants to choose an action 𝑎𝑎𝑖𝑖 such that 𝑓𝑓𝑖𝑖
is positive and large, i.e., player 𝑖𝑖 wants to be kind to player 𝑗𝑗.
247
Fairness vs. Material Well-Being
• Importantly, a player will trade off material well-being against reciprocity: if a
reciprocal response is too costly in terms of material well-being, player 𝑖𝑖 might,
for instance, treat a kind player 𝑗𝑗 in an unkind way if the material gain from
being unkind is sufficiently large.
1
• Recall that the kindness functions are bounded in between −1, . This
2
implies that the relative importance of fairness is diminished if the material
payoffs, which are not bounded, become larger.
• This captures stylized fact (C): Fairness concerns play a bigger role if material
stakes are smaller.
• The model does not pin down the exact relative importance of fairness vs.
material well-being; it just captures the qualitative (i.e., not quantitative)
trade-off.
248
Psychological Nash Equilibrium
• In standard game theory, a player’s utility depends only on the actions chosen by
all players (which determine the material payoffs). In contrast, in “psychological
games,” a player’s utility may also directly depend on the beliefs of players.
• A psychological Nash equilibrium (here: fairness equilibrium) is the analog of a
standard Nash equilibrium for “psychological games” and imposes the additional
condition that all higher-order beliefs (and not only first-order beliefs) must
match actual behavior.
• Definition 3: The pair of strategies 𝑎𝑎1 , 𝑎𝑎2 ∈ (𝑆𝑆1 , 𝑆𝑆2 ) is a fairness equilibrium if,
for 𝑖𝑖 = 1,2, 𝑗𝑗 ≠ 𝑖𝑖,
(1) 𝑎𝑎𝑖𝑖 ∈ arg max 𝑈𝑈𝑖𝑖 (𝑎𝑎, 𝑏𝑏𝑗𝑗 , 𝑐𝑐𝑖𝑖 )
𝑎𝑎∈𝑆𝑆𝑖𝑖
(2) 𝑐𝑐𝑖𝑖 = 𝑏𝑏𝑖𝑖 = 𝑎𝑎𝑖𝑖 .
249
Example of a Fairness Equilibrium (cont., 4)
• Coming back to our earlier example, is the hostile outcome (opera, boxing) a
fairness equilibrium?
• Consistency of beliefs and actions requires:
 𝑐𝑐1 = 𝑏𝑏1 = 𝑎𝑎1 = opera: player 1’s second-order belief 𝑐𝑐1 (indicating what
player 1 believes about player 2’s belief about player 1’s strategy) equals
player 2’s belief 𝑏𝑏1 (indicating what player 2 believes about player 1’s
strategy) and it equals player 1’s strategy 𝑎𝑎1 .
 𝑐𝑐2 = 𝑏𝑏2 = 𝑎𝑎2 = boxing: player 2’s second-order belief 𝑐𝑐2 (indicating what
player 2 believes about player 1’s belief about player 2’s strategy) equals
player 1’s belief 𝑏𝑏2 (indicating what player 1 believes about player 2’s
strategy) and it equals player 2’s strategy 𝑎𝑎2 .
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Example of a Fairness Equilibrium (5)
• Would player 1 have an incentive to deviate from playing “opera”?
• Player 1’s utility from playing opera is 𝑈𝑈 = 0.
• To see this, consider first player 1’s material payoff in this equilibrium, which is
given by 𝜋𝜋1 𝑎𝑎1 = opera, 𝑏𝑏2 = boxing = 0.
• Consider next, player 1’s belief about how kind player 2 is being to him:
• To calculate 𝑓𝑓�2 𝑏𝑏2 = boxing, 𝑐𝑐1 = opera =
𝜋𝜋1 𝑐𝑐1 ,𝑏𝑏2 −𝜋𝜋1𝑒𝑒 𝑐𝑐1
𝜋𝜋1ℎ 𝑐𝑐1 −𝜋𝜋1𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐1
, we must derive its
components:
 Payoff in equilibrium: 𝜋𝜋1 𝑐𝑐1 = opera, 𝑏𝑏2 = boxing = 0
 Highest possible payoff given 𝑐𝑐1 : 𝜋𝜋1ℎ 𝑐𝑐1 = opera = 2𝑋𝑋
 Lowest possible payoff given 𝑐𝑐1 : 𝜋𝜋1𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐1 = opera = 0
251
Example of a Fairness Equilibrium (6)
 Lowest possible payoff among the Pareto-efficient ones given 𝑐𝑐1 :
𝜋𝜋1ℓ 𝑐𝑐1 = opera = 2𝑋𝑋
 Equitable payoff (reference point for fairness evalution):
𝜋𝜋1ℎ 𝑐𝑐1 =opera + 𝜋𝜋1ℓ 𝑐𝑐1 =opera
2𝑋𝑋 +2𝑋𝑋
𝑒𝑒
=
= 2𝑋𝑋
𝜋𝜋1 𝑐𝑐1 = opera =
2
2
• Player 1’s belief about how kind player 2 is being to him is thus given by:
𝑓𝑓�2 𝑏𝑏2 = boxing, 𝑐𝑐1 = opera =
𝜋𝜋1 𝑐𝑐1 , 𝑏𝑏2 − 𝜋𝜋1𝑒𝑒 𝑐𝑐1
𝜋𝜋1ℎ 𝑐𝑐1 − 𝜋𝜋1𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐1
0 − 2𝑋𝑋
=
= −1
2𝑋𝑋 − 0
• Next, we must consider how kind player 1 is to player 2 by choosing 𝑎𝑎1 = opera.
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Example of a Fairness Equilibrium (7)
• Player 1’s kindness (by choosing 𝑎𝑎1 ) to player 2 is given by:
𝑓𝑓1 𝑎𝑎1 , 𝑏𝑏2 ≡
𝜋𝜋2 𝑏𝑏2 , 𝑎𝑎1 − 𝜋𝜋2𝑒𝑒 𝑏𝑏2
𝜋𝜋2ℎ 𝑏𝑏2 − 𝜋𝜋2𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏2
• To calculate𝑓𝑓1 𝑎𝑎1 , 𝑏𝑏2 , we must derive its components:
 Payoff in equilibrium: 𝜋𝜋2 𝑏𝑏2 = boxing, 𝑎𝑎1 = opera = 0
 Highest possible payoff given 𝑏𝑏2 : 𝜋𝜋2ℎ 𝑏𝑏2 = boxing = 2𝑋𝑋
 Lowest possible payoff given 𝑏𝑏2 : 𝜋𝜋2𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏2 = boxing = 0
 Lowest possible payoff among the Pareto-efficient ones given 𝑐𝑐1 :
𝜋𝜋2ℓ 𝑏𝑏2 = boxing = 2𝑋𝑋
 Equitable payoff: 𝜋𝜋1𝑒𝑒 𝑏𝑏2 = boxing =
2𝑋𝑋 +2𝑋𝑋
2
= 2𝑋𝑋
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Example of a Fairness Equilibrium (8)
• Player 1’s kindness to player 2 is thus given by:
𝑓𝑓1 𝑎𝑎1 , 𝑏𝑏2 ≡
𝜋𝜋2 𝑏𝑏2 , 𝑎𝑎1 − 𝜋𝜋2𝑒𝑒 𝑏𝑏2
𝜋𝜋2ℎ 𝑏𝑏2 − 𝜋𝜋2𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏2
0 − 2𝑋𝑋
= −1
=
2𝑋𝑋 − 0
• We can now derive player 1’s utility in equilibrium:
𝑈𝑈1 𝑎𝑎1 = opera, 𝑏𝑏2 = boxing, 𝑐𝑐1 = opera = 𝜋𝜋1 𝑎𝑎1 = opera, 𝑏𝑏2 = boxing +
𝑓𝑓̃2 𝑏𝑏2 = boxing, 𝑐𝑐1 = opera ⋅ 1 + 𝑓𝑓1 𝑎𝑎1 = opera, 𝑏𝑏2 = boxing =
0 + −1 ⋅ 1 + −1 = 0
• What would player 1’s utility be by playing boxing? If his utility is not higher by
deviating to boxing, he does not have an incentive to do so.
254
Example of a Fairness Equilibrium (9)
• By playing boxing, player 1 would increase his monetary payoff and obtain
𝜋𝜋1 𝑎𝑎1 = boxing, 𝑏𝑏2 = boxing = 𝑋𝑋
• How kind would player 1 be to player 2 in this case?
𝜋𝜋2 𝑏𝑏2 , 𝑎𝑎1 − 𝜋𝜋2𝑒𝑒 𝑏𝑏2
2𝑋𝑋 − 2𝑋𝑋
=0
𝑓𝑓1 𝑎𝑎1 = boxing, 𝑏𝑏2 = boxing ≡ ℎ
=
𝑚𝑚𝑚𝑚𝑚𝑚
2𝑋𝑋 − 0
𝜋𝜋2 𝑏𝑏2 − 𝜋𝜋2
𝑏𝑏2
• Player 1 would thus be neither kind nor unkind.
• Player 1’s utility when deviating (from playing opera to playing boxing):
𝑈𝑈1 𝑎𝑎1 = boxing, 𝑏𝑏2 = boxing, 𝑐𝑐1 = opera = 𝜋𝜋1 𝑎𝑎1 = boxing, 𝑏𝑏2 = boxing +
𝑓𝑓̃2 𝑏𝑏2 = boxing, 𝑐𝑐1 = opera ⋅ 1 + 𝑓𝑓1 𝑎𝑎1 = boxing, 𝑏𝑏2 = boxing =
𝑋𝑋 + −1 ⋅ 1 + 0 = 𝑋𝑋 − 1
• Note that we consider a possible deviation from equilibrium, thereby keeping
constant player 2’s actions and beliefs, and thus fairness.
255
Example of a Fairness Equilibrium (10)
• That is, if 𝑋𝑋 < 1, i.e., if the monetary payoffs are relatively small, player 1 does not
want to deviate from playing opera (given his beliefs and second-order beliefs)
 he prefers to forgo money to reciprocate player 2’s unkind intentions.
• Since the game is symmetric, the same reasoning applies to player 2.
• This shows that (opera, boxing) is a fairness equilibrium of the game for sufficiently
small monetary payoffs (𝑋𝑋 < 1).
• By symmetry: (boxing, opera) is also a fairness equilibrium if 𝑋𝑋 < 1.
• Since the kindness functions take on value 0 in case the two players coordinate, both
“classic” Nash equilibria in pure strategies, (opera, opera) and (boxing, boxing), are
also fairness equilibria of the game.
• The interesting implication of the model are the additional equilibria, showing that
fairness motives can explain why the players are willing to impose costly punishment
on each other in equilibrium.
256
Fairness Equilibria in the Prisoner’s Dilemma (1)
• The analysis of the battle-of-the-sexes game exemplified “the dark side of
fairness,” which can lead to hostility in equilibrium.
• By contrast, fairness may also lead each player to sacrifice material payoffs to
help the other player, which we will see in the analysis of the prisoner’s dilemma.
Player 2
Player 1
Cooperate
Defect
Cooperate
4X, 4X
0, 6X
Defect
6X, 0
X, X
257
Fairness Equilibria in the Prisoner’s Dilemma (2)
• There is a unique Nash equilibrium in dominant strategies in this game: (defect,
defect).
• (A dominant strategy is a strategy that is optimal for a player irrespective of the
strategies of other players.)
• Can the outcome (cooperate, cooperate) be sustained as a fairness equilibrium?
• Suppose (cooperate, cooperate) is an equilibrium.
• Consider first player 1. Would he have an incentive to deviate from playing
cooperate?
• Player 1’s utility from playing cooperate is 𝑈𝑈 = 4𝑋𝑋 +
3
.
4
• To see this, consider first player 1’s material payoff in equilibrium, which is given
by 𝜋𝜋1 𝑎𝑎1 = cooperate, 𝑏𝑏2 = cooperate = 4𝑋𝑋.
258
Fairness Equilibria in the Prisoner’s Dilemma (3)
• Consider next, player 1’s belief about how kind player 2 is being to him:
• To calculate 𝑓𝑓�2 𝑏𝑏2 = coop, 𝑐𝑐1 = coop =
𝜋𝜋1 𝑐𝑐1 ,𝑏𝑏2 −𝜋𝜋1𝑒𝑒 𝑐𝑐1
𝜋𝜋1ℎ 𝑐𝑐1 −𝜋𝜋1𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐1
, we must derive its
components:
 Payoff in equilibrium: 𝜋𝜋1 𝑐𝑐1 = coop, 𝑏𝑏2 = coop = 4𝑋𝑋
 Highest possible payoff given 𝑐𝑐1 : 𝜋𝜋1ℎ 𝑐𝑐1 = coop = 4𝑋𝑋
 Lowest possible payoff given 𝑐𝑐1 : 𝜋𝜋1𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐1 = coop = 0
 Lowest possible payoff among the Pareto-efficient ones given 𝑐𝑐1 :
𝜋𝜋1ℓ 𝑐𝑐1 = coop = 0 (note that the payoff pair (0,6X) is Pareto-efficient)
 Equitable payoff:
𝜋𝜋1ℎ 𝑐𝑐1 =coop + 𝜋𝜋1ℓ 𝑐𝑐1 =coop
4𝑋𝑋 +0
𝑒𝑒
𝜋𝜋1 𝑐𝑐1 = coop =
=
= 2𝑋𝑋
2
2
259
Fairness Equilibria in the Prisoner’s Dilemma (4)
• Player 1’s belief about who kind player 2 is being to him is thus given by:
𝑓𝑓�2 𝑏𝑏2 = coop, 𝑐𝑐1 = coop =
𝜋𝜋1 𝑐𝑐1 , 𝑏𝑏2 − 𝜋𝜋1𝑒𝑒 𝑐𝑐1
4𝑋𝑋 − 2𝑋𝑋 1
=
=
4𝑋𝑋 − 0
2
𝜋𝜋1ℎ 𝑐𝑐1 − 𝜋𝜋1𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐1
• Next, we consider how kind player 1 is being to player 2 by choosing 𝑎𝑎1 = coop.
• To calculate𝑓𝑓1 𝑎𝑎1 , 𝑏𝑏2 , we must derive its components:
 Payoff in equilibrium: 𝜋𝜋2 𝑏𝑏2 = coop, 𝑎𝑎1 = coop = 4𝑋𝑋
 Highest possible payoff given 𝑏𝑏2 : 𝜋𝜋2ℎ 𝑏𝑏2 = coop = 4𝑋𝑋
 Lowest possible payoff given 𝑏𝑏2 : 𝜋𝜋2𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏2 = coop = 0
 Lowest possible payoff among the Pareto-efficient ones given 𝑐𝑐1 :
𝜋𝜋2ℓ 𝑏𝑏2 = coop = 0
 Equitable payoff: 𝜋𝜋1𝑒𝑒 𝑏𝑏2 = coop =
4𝑋𝑋 +2𝑋𝑋
2
= 2𝑋𝑋
260
Example of a Fairness Equilibrium (5)
• Player 1’s kindness to player 2 is thus given by:
𝑓𝑓1 𝑎𝑎1 , 𝑏𝑏2 ≡
𝜋𝜋2 𝑏𝑏2 , 𝑎𝑎1 − 𝜋𝜋2𝑒𝑒 𝑏𝑏2
𝜋𝜋2ℎ 𝑏𝑏2 − 𝜋𝜋2𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏2
4𝑋𝑋 − 2𝑋𝑋 1
=
=
4𝑋𝑋 − 0
2
• We can now derive player 1’s utility in equilibrium:
𝑈𝑈1 𝑎𝑎1 , 𝑏𝑏2 , 𝑐𝑐1 = 𝜋𝜋1 𝑎𝑎1 , 𝑏𝑏2 + 𝑓𝑓̃2 𝑏𝑏2 , 𝑐𝑐1 ⋅ 1 + 𝑓𝑓1 𝑎𝑎1 , 𝑏𝑏2
1
1
3
4𝑋𝑋 + ⋅ 1 + = 4𝑋𝑋 +
2
2
4
=
• Would player 1 have an incentive to deviate from playing cooperation?
261
Fairness Equilibria in the Prisoner’s Dilemma (6)
• By playing defect, player 1 would increase his monetary payoff and obtain
𝜋𝜋1 𝑎𝑎1 = defect, 𝑏𝑏2 = coop = 6𝑋𝑋
• How kind would player 1 be to player 2 in this case?
𝜋𝜋2 𝑏𝑏2 , 𝑎𝑎1 − 𝜋𝜋2𝑒𝑒 𝑏𝑏2
0 − 2𝑋𝑋
1
=−
𝑓𝑓1 𝑎𝑎1 = defect, 𝑏𝑏2 = coop ≡ ℎ
=
𝑚𝑚𝑚𝑚𝑚𝑚
4𝑋𝑋 − 0
2
𝜋𝜋2 𝑏𝑏2 − 𝜋𝜋2
𝑏𝑏2
• Player 1 would thus be unkind.
• Player 1’s utility when deviating (from playing cooperation to playing defect):
𝑈𝑈1 𝑎𝑎1 = defect, 𝑏𝑏2 = coop, 𝑐𝑐1 = coop = 𝜋𝜋1 𝑎𝑎1 = defect, 𝑏𝑏2 = coop +
𝑓𝑓̃2 𝑏𝑏2 = coop, 𝑐𝑐1 = coop ⋅ 1 + 𝑓𝑓1 𝑎𝑎1 = defect, 𝑏𝑏2 = coop =
1
1
1
6𝑋𝑋 + ⋅ 1 − = 6𝑋𝑋 +
2
2
4
262
Fairness Equilibria in the Prisoner’s Dilemma (7)
• Deviation from cooperate to defect yields a lower utility for player 1 if
1
3
6𝑋𝑋 + < 4𝑋𝑋 +
4
4
1
𝑋𝑋 <
4
• Since the game is symmetric, the same reasoning applies to player 2.
• The strategy pair (cooperate, cooperate) is thus a fairness equilibrium if the
monetary payoffs are sufficiently small (𝑋𝑋 < 1/4).
• In the fairness equilibrium (cooperate, cooperate), each player wants to help the
other player by playing cooperate, thereby forgoing material gains from defecting.
• It can be shown that the conventional Nash equilibrium (defect, defect) is also a
fairness equilibrium of the game.
263
Reciprocity vs. Pure Altruism
• The concept of pure, or unconditional, altruism cannot explain that both (defect,
defect) and (cooperate, cooperate) can be outcomes of the game.
• To see this, consider the following:
1) If player 1 thought that player 2 was playing cooperate, he would be paying
cooperate if he was willing to give up 2𝑋𝑋 to increase player 2’s payoff by 4𝑋𝑋.
2) If player 1 thought that player 2 was playing defect, he would be paying
cooperate if he was willing to give up 𝑋𝑋 to increase player 2’s payoff by 5𝑋𝑋.
• Hence, if player 1 cooperates in response to cooperation (point 1), he would
cooperate in response to defect (point 2).
• The concept of pure altruism thus cannot explain both equilibria, though both
outcomes seem plausible.
264
Importance of Non-Chosen Strategies (1)
• In the Mini Ultimatum Game, we have seen the importance of non-chosen
strategies (i.e., the respective alternatives to allocation 8:2).
• To see this point in the context of the Rabin-model, consider the following game,
labelled “Prisoner’s Non-Dilemma.”
Player 2
Cooperate
Player 1
Cooperate
4X, 4X
Defect
6X, 0
• Player 2 has no choice but must play cooperate.
265
Importance of Non-Chosen Strategies (2)
• In the Prisoner’s Non-Dilemma, (cooperate, cooperate) is no fairness equilibrium.
• To see this, consider that player 2 cannot be kind (she cannot be unkind either)
because she cannot take an action.
• Since 𝑓𝑓�2 = 0, player 1 will maximize his material well-being and choose to defect.
• It follows that the unique fairness equilibrium (and unique Nash equilibrium) is
given by (defect, cooperate).
• The reason behind (cooperate, cooperate) being no fairness equilibrium in the
Prisoner’s Non-Dilemma is that player 2 does not have “defect” in her strategy
space and thus cannot intentionally forgo choosing defect, thereby being kind to
player 1.
266
Relation of Fairness and Nash Equilibria (1)
• In the examples discussed so far, all Nash-equilibria were also fairness equilibria,
but not all fairness equilibria were Nash-equilibria.
• This is not a general property of the model, as the following “chicken game”
exemplifies:
Player 2
Dare
Chicken
Player 1
Dare
Chicken
-2X, -2X
2X, 0
0, 2X
X, X
• The two Nash-equilibria in pure strategies are (dare, chicken) and (chicken, dare).
• Are these also fairness equilibria of the game?
267
Relation of Fairness and Nash Equilibria (2)
• Consider the Nash-equilibrium (dare, chicken).
• Given this strategy combination, player 1 is unkind to player 2 because he could
choose “chicken” and thereby increase player 2’s payoff from 0 to 𝑋𝑋.
• Player 2 would thus want to be unkind herself (negative reciprocity) and choose
“dare,” provided the cost in terms of material well-being are not too large.
• Hence, for sufficiently small 𝑋𝑋, (dare, chicken) is not a fairness equilibrium of the
game, even though it is a Nash-equilibrium.
• Due to the symmetry of the game, the same applies to (chicken, dare).
• This exemplifies that Nash-equilibria can exist that are not consistent with
fairness motives.
• The paper by Rabin (1993) derives general properties of the model, and discusses
some applications and limitations, all of which is left as optional reading.
268