Supplementary Material for

Social Comparisons and Reference Group Formation: Experimental Evidence By Ian McDonald, Nikos Nikiforakis, Nilss Olekalns and Hugh Sibly

The supplementary material is divided into two sections. The first section presents the extended model of cognitive dissonance for our three-player ultimatum game. The second section includes the experimental instructions.

Section 1: An extended model of cognitive dissonance and reference group formation

1

Introduction to the extended model

In this appendix we show that the conclusions from the model presented in the paper hold in a more general setting where participants can choose any o¤er/MAO in [0; ]. The analysis proceeds as follows: In section 1, the extended version of the model of cognitive dissonance in the three person ultimatum game is developed. The game played by the proposer and responder is shown to be equivalent to a game in which there is a simultaneous choice of the responder’s referent by both players. In section 2, it is shown that, for all Y 2 [0; =2], a pure-strategy equilibrium occurs in which the non-responder is treated as the referent for the responder by both players. In section 3, it is shown that, for low values of Y , a pure-strategy equilibrium may occur in which the proposer is treated as the referent for the responder by both players. In section 4, the mixed-strategy Nash equilibrium, which occurs when there are multiple pure-strategy Nash equilibria, is characterised. When the mixed-strategy equilibrium is played, rejections occur in realisations when the responder chooses the proposer as referent, and the proposer chooses the non-responder as referent. In section 5, it is concluded that the rejection rate increases as Y falls. Therefore, the conclusions in sections 2 to 5 parallells those obtained in the simple model.

2

The Game

Each player chooses who they believe to be the referent for the responder and simultaneously chooses their o¤er/MAO. Participant i’s utility from a share of the surplus (see equation 1 in the paper) is given by: Ui = Xi

ai g(Xi

Ii )

ci

where g(x) is the cognitive dissonance cost experienced by player i when monetary payo¤s di¤er from their perceived entitlement (Ii ), ai is the weighting of this cost in utility, and is an indicator variable taking the value of 1 if player i adopts the proposer as the responder’s referent, and 0 when the non-responder is the referent. It is assumed that g(x) is U-shaped with its minimum point at g(0) = 0. Let Fi be the division of the pie which player i believes the responder is entitled to. Recall that FR = IR and FP = IP . Note that a player’s referent choice is equivalent to the choice of the responder’s entitlement, Fi , so that Fi = Y is equivalent to player i choosing the non-responder as the responder’s referent and Fi = =2 is equivalent to player i choosing the proposer as the responder’s referent. Further: =

0 if Fi = Y 1 if Fi = =2

The set of strategies available to the proposer is sP fZ; FP g where Z is the o¤er. The set of strategies available to the responder is sR fM; FR g where M is the MAO. If the o¤er is accepted, XR = Z and XP = Z. If a rejection occurs, the surplus to be shared between the parties is 0. Hence the entitlement of each participant is 0. Thus, when a rejection occurs, XR = IR = XP = IP = 0.

2.1

The responder’s payo¤

The responder’s utility from the playing the game is: 2

VR (Z; FP ; M; FR ) =

Z

aR g(Z

FR )

if M Z; if M > Z

cR

cR

(1)

where FR 2 fY; =2g. Lemma 1 The minimum o¤er the responder would accept if they adopt referent FR , c(FR ), is de…ned implicity by: M c = aR g(M c M

c(Y ) < Y and 0 < Further M

c(Y ) @M @Y

(2)

FR ) + cR

< 1 for Y 2 (0; =2].

Proof. Equation (2) follows from equation (1). By the implicit function theorem, we c(FR ) with FR = fY; =2g. Observe that can denote the responder’s choice of MAO as M c; Y ) = 0, therefore M c(Y ) must lie VR (Y; FP ; Y; Y ) = Y > 0. By de…nition, VR (Y; FP ; M below Y . Taking the total derivative of (2) gives: dM

aR g 0 (M

FR )(dM

dFR ) = 0

or: 0<

c(Y ) dM dY

=

aR g 0 (M Y ) 1 aR g 0 (M Y )

< 1; as g 0 (M

Y ) < 0 by the convexity of g(:).

c(FR ) is the minimum o¤er that provides Lemma 1 is illustrated in Figure S1. M the responder with non-negative utility. An increase in FR , from FR to F + , shifts the R c(FR ) to M c(F + ). responder’s utility function to the right, increasing the MAO from M R c c Consequently M (Y ) < M ( =2). Note that the MAO is independent of both the o¤er and referent of the proposer, i.e. it is independent of the strategy adopted by the proposer.

2.2

The proposer’s payo¤

The proposer’s utility from playing the game is: VP (Z; FP ; M; FR ) =

Z cP

aP g(FP

Z)

cP

if Z M if Z < M

(3)

De…ne the proposer’s "preferred" o¤er, Z (FP ), as: Z (FP ) = max f0; arg max UP g z

where UP = Z aP g(FP Z) cP . The proposer’s "preferred" o¤er is the one that the proposer would set if guaranteed of its acceptance. De…ne FeP by aP g 0 (FeP ) = 1. An interpretation of FeP is given below. The following lemma obtains some useful properties of the proposer’s prefererred o¤er. Lemma 2 (i) For FP 6 FeP , Z (FP ) = 0. For FP > FeP the preferred o¤er satis…es: aP g 0 (FP

Z )=1

(ii) Z (FP ) < FP . When FP = Y > FeP then: (iii) @Z@Y(Y ) = 1, (iv) 3

@Z (Y ) @Y

>

c(Y ) @M : @Y

(4)

Proof. (i) Note that dUP =dZ = 1 + aP g 0 (FP Z); and that dUP =dZ = 0 when (4) holds. Taking the total derivative of (4), dZ =dFP > 0, and, by de…nition, Z = 0 if FP = FeP . Consequently, Z (FP ) = 0 for all FP 6 FeP and Z (FP ) satis…es (4) for all FP > FeP . (ii) From (4), we require g 0 (FP Z ) > 0 for Y > FeP . The convexity of g(:) means that FP Z > 0. (iii) By taking the total derivative of (4), we obtain aP g 00 (FP Z )[dFP dZ ] = 0. For this to be the case, it must be that [dFP dZ ] = 0, thus dZ (FP )=dFP = 1: (iv) Follows from lemma 1 and lemma 2(iii). The preferred o¤er is illustrated in Figure S2. The preferred o¤er is less than the proposer’s entitlement, because the proposer gains materially from a low o¤er. As shown in Figure S2, an increase in FP , from FP to FP+ , shifts the proposers utility function to the right, increasing the value of FP for which the maximum of occurs. The preferred o¤er maximises UP . Note if FP = FeP then Z = 0 maximises UP . Any value of FP < FeP causes the proposer to make an o¤er of Z = 0. FeP is thus the maximum value of Fp that results in a pre¤ered o¤er of 0. e P ) as the maximum o¤er that earns the proposer non-negative utility, De…ne Z(F e VP 0. Z(FP ) is illustrated in …gure S2. Then: e P) > M c(Y ) for FP 2 fY; =2g. Lemma 3 Z(F

Proof. First, suppose that FP = Y . Note that when Z = Y , then from equation (3), e ), it follows that Z(Y e )>Y. VP (Y; Y; M; FR ) = Y > 0. Hence, by the de…nition of Z(Y e =2) > Z(Y e ); as Z(F e P ) is increasing in FP . Consequently, Z( e =2) > Y . Now note that Z( c(Y ) < Y , hence, Z(F e P) > M c(Y ) for FP 2 fY; =2g. From lemma 1, M

Lemma 3 shows that the proposer will be able to make an o¤er that generates them positive utility if the responder adopts the non-responder as referent. The proposer’s o¤er c. maximises their payo¤ subject to the responder’s M There are three cases to consider when specifying the proposer’s o¤er. These three cases are illustrated in Figure S3, which shows the proposer’s utility function and three c1 ; M c2 ; M c3 ). In the …rst case M c(FR ) < Z (FP ), consequently the propossible MAOs (M poser’s o¤er is Zb = Z (FP ). This case is illustrated in Figure S3 by the example with c(FR ) = M c1 . If Z (FP ) c(FR ) e P ) the proposer’s o¤er is Zb = M c(FR ). This M M Z(F c(FR ) = M c2 . If Z (FP ) Z(F e P) case is illustrated in Figure S3 by the example with M c(FR ) the proposer’s o¤er is assumed to be Zb = Z(F e P ). This case is illustrated in Figure M c(FR ) = M c3 . Thus the proposer’s o¤er satis…es: S3 by the example with M

2.3

8 < Z (FP ); b ^ (F ); Z(FP ; FR ) = M : e R Z(FP );

if if if

^ (FR ) < Z (FP ) M ^ (FR ) Z (FP ) M e P) Z (FP ) Z(F

e P) Z(F ^ (FR ) M

(5)

The game as a choice of referents

The above analysis shows that the proposer’s o¤er and the responder’s MAO depend on the participants’perceived entitlements. In this section, we show how the game can be interpreted and analysed as one in which players simultaneously choose their referents.

4

In the above analysis, we showed the responder’s MAO was independent of the strategy adopted by the proposer. However, the responder’s utility from playing the game is dependent on both players’perceived entitlement, VbR (FP ; FR ), in the following way: VbR (FP ; FR )

b P ; FR ); FP ; M c(FR ); FR ) = V (Z(F (R b P ; FR ) aR g(Z(F b P ; FR ) FR ) Z(F = cR

cR

if if

b P ; FR ) M c(FR ) Z(F b P ; FR ) < M c(FR ) Z(F

Similarly, the proposer’s utility from the game is dependent on the player’s perceived entitlement, VbP (FP ; FR ), as follows: VbP (FP ; FR )

b P ; FR ); FP ; M c(FR ); FR ) = V (Z(F (P b P ; FR ) aP g(FP Z(F b P ; FR )) Z(F = cP

cP

if if

b P ; FR ) M c(FR ) Z(F b P ; FR ) < M c(FR ) Z(F

If the proposer chooses the non-responder as referent, they adopt the strategy comb Y ); Y ). Similarly, if the responder adopts the non-responder as bination sYP (Z(Y; c(Y ); Y ). On the other hand, if the proposer adopts themreferent they choose sYR (M =2 b =2; =2); =2), while selves as referent they choose the strategy comination sP (Z( if the responder adopts the proposer as referent they choose the strategy comination =2 c( =2); =2). The game may now be represented in the following strategic sR (M form: Table 1 Responder =2

Proposer

sYP =2 sP

sYR sR b b b VP (Y; Y ); VR (Y; Y ) VP (Y; =2); VbR (Y; =2) VbP ( =2; Y ); VbR ( =2; Y ) VbP ( =2; =2); VbR ( =2; =2)

Below we show that, depending on the value of Y , the game in Table 1 may be either a game with a dominant strategy equilibrium or may be a battle-of-the-sexes coordination game.

3

Pure-strategy equilibrium with the non-responder as a common referent (FP = FR = Y )

b Y ); Y ) and sY c(Y ); Y )). Consider the strategy combination (sYP ; sYR ). (Recall sYP (Z(Y; (M R In this strategy combination both players are treating the non-responder as the referent, and setting the o¤er/MAO accordingly. Before demonstrating that (sYP ; sYR ) is an equilibrium strategy combination, we show some useful preliminary results. Assume the proposer and responder adopt the non-responder as referent. Equation c(Y ), where Y is the value of Y for (2) identi…es the responder’s MAO. De…ne Z (Y ) = M 5

which the proposer’s preferred o¤er equals the responder’s MAO. The proposer’s o¤er is found from (5) as: Z (Y ) if Y >Y c(Y ) if Y Y M

b Y)= Z(Y;

(6)

b Y ) is illustrated in Figure S4. Lemmas 1 and 2 show that Z (Y ) is The curve Z(Y; c(Y ). Thus the proposer o¤ers M c(Y ) for relatively low Y , and Z (Y ) for steeper than M relatively high Y . The proposer’s utility from (sYP ; sYR ) is: Z (Y ) c(Y ) M

VbP (Y; Y ) =

Z (Y )) if Y >Y c(Y )) if Y Y M

aP g(Y aP g(Y

The responder’s utility from (sYP ; sYR ), is: Z (Y ) 0

VbR (Y; Y ) =

aR g(Z (Y )

Y)

(7)

if Y >Y if Y Y

(8)

From (6), (7) and (8) we have the following result which will be useful below: b

) Lemma 4 For all Y 6=Y : 0< @ Z(Y;Y @Y

1,

@ VbP (Y;Y ) @Y

< 0 and

@ VbR (Y;Y ) @Y

0.

Proof. Follows from (6), (7), (8) and lemma 2. Lemma 4 states that as the payment to the non-responder increases, the proposer’s o¤er increases their utility decreases , and the responder’s utility increases. Lemma 4 is used to prove the …rst of our propositions: Proposition 1 (sYP ; sYR ) is a pure strategy Nash equilibrium for 0 < Y <

=2 .

Proof. First we demonstrate that the responder does not increase their utility by unilaterally deviating from (sYP ; sYR ). Changing strategies yields a payo¤:

VbR (Y; =2) =

(

Z (Y ) cR

aR g(Z (Y )

=2)

cR

c( =2) if Z (Y ) M c( =2) if Z (Y ) < M

The responder will not unilaterally deviate from (sYP ; sYR ) if VbR (Y; =2) < VbR (Y; Y ). c( =2) the responder’s payo¤ falls from 0 to cR We consider two cases: (i) If Z (Y ) < M (as the responder su¤ers the cognitive dissonance of adopting the proposer as referent). In this case the responder does not have an incentive to unilaterally deviate from (sYP ; sYR ). c( =2), in order for the responder to adopt the proposer as referent, it (ii) If Z (Y ) M would be necessary that: Z (Y )

aR g(Z (Y )

Y ) < Z (Y )

aR g(Z (Y )

=2)

cR

or: cR < aR [g(Z (Y )

Y)

6

g(Z (Y )

=2)]

(9)

b Y ) < Y < =2, g(Z (Y; Y ) Y ) < g(Z (Y; Y ) Since Z(Y; =2). Thus the RHS of (9) is negative. This implies that cR < 0, which contradicts the assumption of the model. Hence the responder does not have an incentive to unilaterally deviate from (sYP ; sYR ). Now it is demonstrated that the proposer cannot increase their utility by unilaterally deviating from (sYP ; sYR ) by adopting the proposer as referent, i.e., by changing strategy b Y ); Y ) to s =2 (Z( b =2; Y ); =2) If the proposer adopted themselves from sYP (Z(Y; P as the referent, from (5), they would make the following o¤er: ( c(Y )) Z ( =2) if Z ( =2) > M b =2; Y ) = Z( c(Y ) c(Y ) < Z( e =2): M if Z ( =2) M c(Y ) > Z( e =2), as M c(Y ) < Y < Z(Y e ) < Z( e =2). Note that it is not possible for M Hence, the third case of equation (5) is not of relevance in this case. For the proposer to unilaterally deviate from (sYP ; sYR ) would require: VbP (Y; Y ) < VbP ( =2; Y )

or: b Y) Z(Y;

aP g(Y

b Y )) < Z(Y;

b =2; Y ) Z(

aP g( =2

b =2; Y )) Z(

Z ( =2)

aP g( =2

Z ( =2))

g( =2

Z ( =2))]

cP :

b =2; Y ) = M c(Y ) then Z(Y; b Y) = M c(Y ). In this case the From lemma 2(iii) if Z( inequality cannot hold as it would require cP < 0, which contradicts the assumptions of b =2; Y ) = Z ( =2) > M c(Y ). For the the model. Now consider the case in which Z( Y Y proposer to unilaterally deviate from (sP ; sR ) would require:

or:

b Y) Z(Y;

aP g(Y

b Y) cP < [Z(Y;

b Y )) < Z(Y;

Z ( =2)] + aP [g(Y

b Y )) Z(Y;

cP (10)

b Y ), hence the …rst term on the RHS of (10) We know from above that Z ( =2) > Z(Y; c(Y ) is negative. From lemma 2, Y Z (Y ) = =2 Z ( =2); and from lemma 1, Y M b Y )) Y Z (Y ) < =2 Z ( =2). Consequently, g(Y Z(Y; g( =2 Z ( =2)), so the second term on the RHS of (10) is non-positive. This implies that cP < 0, which is a contradiction. Hence, the proposer does not unilaterally deviate from (sYP ; sYR ).

4

Pure-strategy equilibrium with the proposer as the common referent (FP = FR = =2) =2

=2

We now show that (sP ; sR ) is a pure-strategy Nash equilibrium for some parameter =2 b =2; =2); =2) and s =2 c( =2); =2). In this constellations, where sP (Z( (M R case, both players are treating the proposer as the referent, and setting the o¤er/MAO accordingly. =2 =2 The proposer’s o¤er given (sP ; sR ) can be written as: ( c( =2) Z ( =2) if Z ( =2) > M b =2; =2) = Z( c( =2) if Z ( =2) M c( =2) M 7

=2

=2

The proposer’s utility from (sP ; sR ) is given by:

VbP ( =2; =2) =

(

Z ( =2) c( =2) M

aP g( =2 aP g( =2

=2

Z ( =2)) c( =2)) M

c( =2) if Z ( =2) > M c( =2) if Z ( =2) M

cP cP

=2

The responder’s utility from (sP ; sR ) is given by:

VbR ( =2; =2) =

(

Z ( =2) cR

aR g(Z ( =2)

=2)

cR

Let Y E = min(YPE ; YRE ), where YPE is de…ned by: cP

c( =2) M

c( =2)) M

Z ( =2) + aP [g(YPE

and where YRE is de…ned by: cR

b =2; =2) aR [g(Z(

YRE )

c( =2) if Z ( =2) > M c( =2) if Z ( =2) M g( =2

b =2; =2) g(Z(

(11)

Z ( =2))]

(12)

=2)]

YPE is the maximum value of Y for which the proposer has no incentive to deviate =2 =2 from (sP ; sR ): Similarly YRE is the maximum value of Y for which the responder has no =2 =2 incentive to deviate from (sP ; sR ). Consequently, as shown in the following proposition, =2 =2 Y E is the maximum value of Y for which (sP ; sR ) is an equilibrium. =2

=2

Proposition 2 If 0 Y Y E the strategy combination (sP ; sR ) is an equilibrium. =2 =2 If either 0 Y E < Y =2 or Y E < 0 the strategy combination (sP ; sR ) is not an equilibrium. Proof. To demonstrate the proposition, we …rst consider if and when the proposer =2 =2 has an incentive to deviate from (sP ; sR ). If the proposer unilaterally deviates form =2 =2 (sP ; sR ), from equation (5) their o¤er would be: 8 c( =2)) > if Z (Y ) > M < Z (Y ) b c( =2) if Z (Y ) M c(Y ) Z(Y e ) Z(Y; =2) = M > : Z(Y e ) e ) M c( =2) if Z (Y ) Z(Y =2

=2

Thus the proposer’s utility from unilaterally deviating from (sP ; sR ) is: 8 c( =2) > Z (Y ) aP g(Y Z (Y )) if Z (Y ) > M < c( =2) aP g(Y M c( =2)) if Z (Y ) M c(Y ) < Z(Y e ) VbP (Y; =2) = M > : 0 e ) M c( =2) if Z (Y ) Z(Y

For the proposer to lower their utility by unilaterally deviating would require VbP (Y; =2) < VbP ( =2; =2). This is equivalent to requiring: Z(Y; =2)

aP g(Y

Z(Y; =2)) <

Z ( =2)

8

aP g( =2

Z ( =2))

cP (13)

c( =2)): where Z(Y; =2) = max(Z (Y ); M c( =2). The determination of Y is illustrated in Figure S4. Y De…ne Y by Z (Y ) M is the minimum value of Y for which the proposer’s preferred o¤er when they choose the non-responder as referent, Z (Y ), is greater than the MAO the responder chooses when c( =2). Then Z(Y; =2) = M c( =2) for all Y < Y they treat the proposer as referent, M and Z(Y; =2) = Z (Y ) for all Y < Y < =2. Equation (13) can be rewritten as: cP < [Z(Y; =2)

Z ( =2)] + aP [g(Y

Z(Y; =2))

g( =2

(14)

Z ( =2))]

From lemma 2(iii), Z ( =2) Z(Y; =2), hence the …rst term on the RHS of (14) is negative. Similarly, from lemma 2(iii), g(Y Z (Y )) = g( =2 Z ( =2)), so the second term is zero for Y < Y < =2. This implies that cP < 0 for Y < Y < =2, which is a contradiction of the assumptions of the model. Hence, the proposer does not lower his utility, as assumed in (13), by adopting the non-responder as the referent when Y < Y < =2 =2 =2. This implies that the proposer has a incentive to unilaterally deviate from (sP ; sR ) for Y < Y < =2. c( =2) > g( =2 Now consider the case 0 < Y < Y . It is possible that g(Y M c( =2) Z ( =2) + Z ( =2)) for su¢ ciently low Y . In this event it is possible that M c( =2)) g( =2 Z ( =2))] > cP > 0. In this case, from (14), the proposer aP [g(Y M =2 =2 can lower their utility by unilaterally deviating from (sP ; sR ). To show this, recall from (11) the de…nition of YPE , and note that the LHS of the above equation is decreasing in Y . =2 The proposer will not deviate from sP when 0 Y YPE . The strategy combination =2 =2 =2. The intuition for this conclusion (sP ; sR ) is not an equilibrium for YPE < Y is illustrated in Figure S4. If the proposer adopts sYP then for low Y , for example YL , there is a relatively large gap between the proposer’s entitilement, =2, and the o¤er they c( =2). This di¤erence creates cognitive dissonance, which may outweight the make M c( =2) Z( b =2; =2), by changing strategies. Finally, note increased material payo¤, M =2 =2 that the strategy combination (sP ; sR ) is not an equilibrium if YPE < 0 for a particular proposer-responder combination, as there is not value of Y for which the proposer does =2 =2 not have an incentive to deviate from (sP ; sR ). We now turn to demonstrating that there are some parameter values for which the responder does not increase their utility by unilaterally deviating the strategy combination =2 =2 b =2; =2) > M c(Y ) for all Y , if the responder plays sY they receive (sP ; sR ). As Z( R utility: b =2; =2) aR g(Z( b =2; =2) Y ) VbR ( =2; Y ) = Z( =2

=2

For the responder to unilaterally deviate from (sP ; sR ) would require:

b =2; =2) Z( or:

b =2; =2) aR g(Z(

b =2; =2) Y ) < Z(

b =2; =2) cR < aR [g(Z(

Y)

b =2; =2) aR g(Z(

b =2; =2) g(Z(

=2)

cR

=2)]:

The RHS of the above inequality is decreasing in Y . Utilising equation (12), it is seen that if YRE 0 the responder will not deviate from adopting the proposer as referent =2 =2 for all 0 < Y YRE .The strategy combination (sP ; sR ) is not an equilibrium for =2 =2 0 YRE < Y =2, as the responder has an incentive to deviate from (sP ; sR ). 9

=2

=2

Consequently the strategy combination (sP ; sR ) is not an equilibrium for any Y if YRE < 0. Proposition 2 shows that if Y is su¢ ciently low, that is, if the payment to the nonresponder is su¢ ciently small, ignoring the non-responder and adopting the proposer as a referent can be an equilibrium. The intuition is the same as that given for the simple model in the paper.

5

Mixed-stragegy Nash Equilibrium =2

In the mixed-strategy equilibrium the responder plays sYR with probability qR and sY =2 with a probability 1 qR . The proposer plays sYP with a probability qP and sP with a probability 1 qP . Proposition 3 Suppose 0 Y Y E and participants play the mixed strategy. The probability the responder plays sYR decreases with Y and the probability the proposer plays sYP increases with Y . Proof. The probability the responder plays sYR in the mixed-strategy Nash equilibrium, qR , must satisfy: qR VbP (Y; Y ) + (1

Thus:

qR )VbP (Y; =2) = qR VbP ( =2; Y ) + (1

qR )VbP ( =2; =2)

VbP ( =2; =2) VbP (Y; =2) VbP (Y; Y ) VbP (Y; =2) VbP ( =2; Y ) + VbP ( =2; =2) From the above analysis, we know that: b Y ) aP g(Y Z(Y; b Y )); VbP (Y; Y ) = Z(Y; VbP ( =2; =2) = VbP ( =2; Y ) = Z ( =2; =2) aP g( =2 Z ( =2; =2) cP ; and ( c( =2) Z (Y ) ap g(Y Z (Y )) if Z (Y ) M Vbp (Y; =2) = c( =2): 0 if Z (Y ) < M c( =2); then Z(Y; b Y ) = Z (Y ), as M c( =2) > M c(Y ). In this case, If Z (Y ) M Y < Y < =2 and, as shown above, a mixed-strategy Nash equilibrium does not exist. Thus: Z ( =2; =2) aP g( =2 Z ( =2; =2) cP qR = (15) b Y ) aP g(Y Z(Y; b Y )) Z(Y; qR =

Di¤erentiation of (15) shows that:

@qR = @Y

qR b Y) Z(Y;

aP g(Y

b Y )) Z(Y;

(

b Y) @ Z(Y; + aP g 0 (Y @Y

b Y )) 1 Z(Y;

c @Z(Y; Y) @Y

b ) b Y ) > 0, so g 0 (Y Z(Y; b Y )) > 0. As 0 < @ Z(Y;Y Note that Y Z(Y; 1, the above @Y @qR equation demonstrates that @Y > 0. The probability the proposer plays sYP in the mixed strategy Nash equilibrium, qP , satis…es:

qP VbR (Y; Y ) + (1

qP )VbR ( =2; Y ) = qP VbR (Y; =2) + (1 10

qP )VbR ( =2; =2)

!)

Hence: qP =

VbR (Y; Y )

VbR ( =2; =2) VbR ( =2; Y ) VbR ( =2; Y ) VbR (Y; =2) + VbR ( =2; =2)

From the above analysis, we know that: b Y ) aR g(Z(Y; b Y ) Y ); VbR (Y; Y ) = Z(Y;

b =2; =2) aR g(Z( b =2; =2) VbR ( =2; =2) = Z( =2) cR ; b =2; =2) aR g(Z( b =2; =2) Y ); and VbR ( =2; Y ) = ( Z( b ) aR g(Z(Y b ) Y ) cR if Z (Y ) M c( =2) Z(Y VbR (Y; =2) = c( =2): cR if Z (Y ) < M

As noted above, a mixed-strategy equilibrium cannot occur for Z (Y ) Hence, in the mixed-strategy Nash equilibrium: qP =

b =2; =2) Y ) g(Z( b =2; =2) aR [g(Z( =2)] cR b b b Z(Y; Y ) + aR [g(Z( =2; =2) Y ) g(Z(Y; Y ) Y ) g(Z ( =2; =2)

b =2; =2) Y ) g(Z( b =2; =2) Write (Y ) = aR [g(Z( b Y ) Y ). It is readily shown that 0 (Y ) < 0 and aR g(Z(Y; qP =

Thus:

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@qP = @Y

0

(Y ) cR (Y ) + (Y )

0 (Y )[ (Y ) + cR ] (Y )[ (Y ) [ (Y ) + (Y )]2

c( =2). M =2)]

b Y) =2)] and (Y ) = Z(Y; 0 (Y )>0. Then:

cR ]

< 0:

Rejections and Y

Proposition 4 (i) Suppose the participants play the mixed strategy. Then rejections =2 occur when the realised strategy combination is (sYP ; sR ). (ii) Suppose the participants play the mixed strategy.The rejection rate decreases with Y. c( =2). However this is simply the Proof. (i) A rejection occurs when Z (Y; Y ) < M E requirement that Y < YP , which in turn is the requirement for the implementation of a P mixed strategy equilibrium.(ii) The rejection rate is given by qP (1 qR ). As @q < 0 and @Y @qR > 0, an increase in Y decreases the rejection rate. @Y Proposition 4 shows that rejections will arise due to a coordination failure in the mixed strategy Nash equilibrium. As with the simplied model presented in the paper, an increase in the payment to the non-responder increases the rejection rate.

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Responder Utility, UR(FR)=Z-aRg(Z-FR)-λcR

UR(FR)

UR(FR+)

Offers, Z ^ ) ^ + M(F R M(FR) Figure S1: Determination of MAO where FR
Proposer Utility, UP(FP)=Π-Z-aPg(FP-Z)-λcP

UP(FP+)

UP(FP) ~

UP(FP)

~

Z*(FP)=0

Z*(FP)

Z*(FP+)

~

Z(FP+) ~

Z(FP)

Offers, Z

Figure S2: Determination of the proposer’s offer where FP
Proposer Utility

UP(FP)

Z*(FP) ^ M1

~

Z(FP+) ^ M2

Figure S3: Determination of the proposer’s offer.

Offers, Z ^ M3

Offer/MAO Z*(Π/2) ^ Z(Y) ^ M(Π/2) ^ M(Y) Z*(Y) ^

Y

Y

Π/2

Figure S4: The proposer’s offer.

Y

Section 2: Sample Experimental Instructions

Instructions    These are the instructions for treatment T5. Instructions for the other treatments were appropriately adjusted.   

You are now taking part in an economic experiment. If you read the following instructions carefully, you can,  depending  on  your  decisions  and  of  those  made  by  the  others,  earn  a  considerable  amount  of  money.  It  is  therefore important that you take your time to understand the instructions.     The  instructions  which  we  have  distributed  to  you  are  for  your  private  information.  Please  do  not  communicate with the other participants during the experiment. Should you have any questions please ask us.       During the experiment we shall not speak of Dollars, but of Experimental Monetary Units (EMU). Your entire  earnings will be calculated in EMUs. At the end of the experiment the total amount of EMUs you have earned  will be converted to Australian Dollars at the rate of 1 EMU = 40 cents and will be immediately paid to you in  cash. Payments will be private and none of the other participants will know how much money you earned in  the experiment. Every participant in the experiment will also receive a daily zone 1 concession MetCard.     At the  beginning of  the experiment the participants will be randomly divided into groups of three.   You will  therefore be in a group with two other participants. The experiment is divided in two stages.     The effort stage  The first stage is common for all participants and we will refer to it as the ‘effort stage’. In the effort stage, all  participants  will  be  given  a  task  that  will  determine  the  role  that  they  will  play  in  their  group  during  the  experiment.  The  task  in  the  effort  stage  is  the  same  for  everyone.  You  will  be  presented  with  a  number  of  words  and  your  task  is  to  code  these  words  by  substituting  the  letters  of  the  alphabet  with  numbers  using  Table 1 on the last page. The effort stage decision screen is seen in Figure 1.     Example: You are given the word FLAT. The letters in Table X show that F=6, L=3, A=8, and T=19.    Once  you  code  a  word  correctly,  the  computer  will  prompt  you  with  another  word  to  encode.  Once  you  encode that word, you will be given another word and so on. This process will continue for 7 minutes (420  seconds). All group members will be given the same words to encode in the same sequence.     Allocation of roles  There are three roles in this experiment: Proposer, Responder and Non‐Responder. The role of the Proposer is  allocated  based  on  the  relative  performance  of  the  individuals  in  your  group.  The  person  who  encodes  the  highest number of words in will be assigned the role of a Proposer. The participants with the second and third  highest number of encoded words in their group will be randomly assigned the role of either the Responder or  the  Non‐Responder.  If  two  or  more  participants  tie  in  the  first  place,  the  computer  will  determine  the  roles  randomly.     At the end of the effort stage you will be informed about whether you encoded the highest number of words  or not. If you have not encoded the highest number of words there will be a random draw that will determine  whether you will have the role of the Responder or the Non‐responder. You will not be informed about the  exact number of words that each group member encoded.    

 

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  Figure 1    The decision stage  The task of each participant in the decision stage depends on the role they are assigned. The Non‐Responder  has no decision to make in this stage and will receive 5 EMU at the end of the experiment. The Proposer will  be given an endowment of 100 EMU. S/he must then propose a division of the 100 EMU by making an offer to  the Responder. The amount can be any integer number from 0 to 100 (inclusive). The Responder can accept or  reject the offer. If the offer is accepted then the suggested division is implemented. If the offer is rejected then  both Proposer and Responder receive 0 EMU. Note that Responders will have to make a decision about the  offer they are willing to accept before they see the Proposer’s actual offer. To do this the Responder will be  prompted to state the minimum amount s/he is willing to accept from the Proposer.     If  the  Proposer’s  offer  is  an  amount  higher  or  equal  to  the  Responder’s  minimum  stated  amount,  then  the  Proposer’s offer will be accepted. The  Responder  will receive the offer and  the Proposer will earn 100 EMU  minus his/her offer. If the Proposer’s offer is lower than the Responder’s minimum stated amount, then the  offer will be rejected. In that case, both the Proposer and the Responder will receive 0 EMU. The earnings of  the Non‐Responder are independent of the decisions of the Proposer and the Responder and equal to 5 EMU.     Each decision will be made only once.     Note that if the minimum offer that the Responder is willing to accept is greater than the Proposers offer then  both the Proposer and the Responder will have zero earnings from the experiment. Therefore, make sure you  take your time to make your decisions.    Control Questions   Please answer the following questions. If you have any questions or have answered all the questions, please  raise your hand and one of the experimenters will come to you.    1. What does the effort stage determine? (Tick the correct answer)  □ the roles of each participant    □ who will have the role of the Proposer  □ who will have the role of the Responder  □ who will have the role of the Non‐Responder    2. The role of the Responder is (Tick the correct answer) 

 

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□ assigned randomly to one of the participants who ranked 2nd and 3rd in the effort stage   □ assigned to the person who ranked 2nd in the effort stage  □ assigned to the person who ranked 3rd in the effort stage    3. Assume that the Proposer offers an amount X to the Responder which is larger than the Responder’s  minimum stated amount. How much will   (i) the Proposer earn?  ..................................................  (ii) the Responder earn?  ..............................................  (iii) the Non‐Responder earn? ……………………………………    4. Assume that the Proposer offers an amount X to the Responder which, however, is smaller than the  Responder’s minimum stated amount. How much will   (i) the Proposer earn?  ..................................................  (ii) the Responder earn?  ..............................................  (iii) the Non‐Responder earn? ……………………………………    Table 1  Letters  Numbers  A  8  B  12  C  14  D  10  E  9  F  6  G  24  H  22  I  7  J  5  K  11  L  3  M  18  N  1  O  21  P  16  Q  23  R  2  S  13  T  19  U  25  V  4  W  26  X  17  Y  20  Z  15    

 

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