Supplementary Materials for webpage

APPENDIX A

INSTRUCTIONS

Thank you for participating in our experiment. You will receive $5 for showing
up on time, plus you will receive your earnings from the choices made in the session.

There will be 25 periods. In each period, each person will be matched with one
other person. The person with whom you are matched will be randomly re-drawn after
every period. You are paired anonymously, which means that you will never learn the
identity of the other person in any of the periods.

One person will have the role of ROW and the other person will have the role of
COLUMN. Your role will also be randomly re-drawn in each period, so that sometimes
you will have the role ROW and sometimes you will have the role of COLUMN.

Here is the basic game:

COLUMN
Left Right

ROW Up

Down

40, 52

8,60

52, 8

28,24

The ROW and COLUMN players make choices simultaneously . The ROW player
chooses Up or Down; the COLUMN player chooses Left or Right.

The 1st number in each cell refers to the payoff (in cents) for the ROW player, while
the 2 nd number in each cell refers to the payoff (in cents) for the COLUMN player.

Thus, for example, if ROW chooses Up and COLUMN chooses Left, the ROW player
would receive 40 and the COLUMN player would receive 52.

However, before these game choices are made, ROW may choose a binding amount to

be paid {transferred) by him or her to COLUMN if and only if COLUMN chooses Left.
COLUMN (at the same time) may offer a binding amount to be transferred to ROW if
and only if ROW chooses Up. These amounts must be non-negative integers.

The amounts that you each choose will be communicated to each of you prior to your
choices in the game above. You will then make your game choice (Up or Down if you
are ROW, or Left or Right if you are COLUMN). You will then learn your payoff for the
period, from which you can infer the game choice made by the person with whom you
are paired.

This completes one period of play. We'll do 25 periods and pay people individually and
privately.

FURTHER EXPLANATION

Offers to pay money contingent on the other person choosing Up (or Left, if the other
person is a COLUMN player) have the effect of changing the payoff matrix. Note that
whatever amount you state will be transferred to the other person if he or she plays Up as
a ROW player or Left as a COLUMN player; this money will be transferred regardless
of your game choice.

Suppose, for example, that ROW offers to pay $x to COLUMN if COLUMN plays Left
and COLUMN offers (independently and simultaneously) to pay $y to ROW if ROW
plays Up. Then the payoff matrix becomes:

COLUMN

Left

ROW

Up
Down

Right

40 + y - x, 52 + x - y

8 + y, 60 - y

52-x,8 + x

28,24

We explain the 4 possible outcomes below. Remember, the values of x and y are always
determined by the ROW and COLUMN players, respectively, before making game
choices.

1) If ROW chooses Up and COLUMN chooses Left, then ROW must pay x units to
COLUMN and COLUMN must pay y units to ROW. Thus, ROW would receive
40 + y - x and COLUMN would receive 52+x-y.

2) If ROW chooses Up and COLUMN chooses Right, then COLUMN must pay y
units to ROW, but ROW pays nothing to COLUMN (because COLUMN did not
choose Left). Thus, ROW would receive 8 + y and COLUMN would receive 60 -

y-

3) If ROW chooses Down and COLUMN chooses Left, then ROW must pay x units
to ROW, but COLUMN pays nothing to ROW (because ROW did not choose
Up). Thus, ROW would receive 52 -x and COLUMN would receive 8 + x.

4) If ROW chooses Down and COLUMN chooses Right, then neither player pays
the other anything. Thus, ROW would receive 28 and COLUMN would receive
24.

We don't wish to illustrate this with an example with realistic numbers, as this could bias
your behavior. However, we can use an example where x = 999 and y = 1000. (We don't
expect anyone to choose these values for x and y.) In this case, the payoff matrix
becomes:

ROW

Up
Down

COLUMN
Left Right

41, 51

1008, -940

-947, 1007

28,24

We encourage people to work out scenarios on paper, drawing a game matrix for each
possibility.

Are there any questions? Please feel free to ask, by raising your hand.

APPENDIX B

Transfer-pair regions consistent with (C,C) being a subgame-perfect action pair

Game 1

(8,20) (16,20)

H 2

(8,12) (16,12)
Hi

Game 2

(8,16) (16,16)

H 2

(8,8) (16,8)
Hi

Game 3

(16,24) (28,24)

(8,8) (20,8)
Hi

APPENDIX C

Determinants of cooperation
Random-effects probit estimates in NE region

Game 1:
Row

Game 1:
Column

Game 2:
Row

Game 2:
Column

Game 3:
Row

Game 3:
Column

Would Pay

-0.03

-0.013

0.056

-0.011

-0.013

0.029

(0.019)

(0.049)

(0.056)

(0.036)

(0.029)

(0.026)

Would Receive

0.102*

0.151***

0.113**

0.012

0.172***

0.076***

(0.054)

(0.055)

(0.050)

(0.051)

(0.033)

(0.022)

NE Border

-0.562

-1 359***

-0.990***

-1.087***

-0.067

-0.45

(0.371)

(0.386)

(0.344)

(0.326)

(0.280)

(0.288)

Equal Transfers

0.935*

0.054

0.432

-0.435

0.711**

0.522*

(0.535)

(0.556)

(0.362)

(0.486)

(0.315)

(0.301)

Final Payments are Closer

-0.262

-0.368

1.129***

-1.060**

0.425*

0.725***

(0.353)

(0.372)

(0.393)

(0.438)

(0.249)

(0.247)

Constant

-0.185

-0.349

-1.702*

2.055**

-2.085***

-1.671***

(0.939)

(1.304)

(0.946)

(0.986)

(0.636)

(0.578)

Observations

228

228

298

298

294

294

Number of Subjects

31

32

31

32

32

32

Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%

APPENDIX D

Determinants of mutual cooperation
Random-effects probit with one way subject error terms
and marginal-effects estimates in SPE region

Game 1

Game 2

Game 3

All Games

All Games:
Marginal

NE Border

-2.138***

-1.275***

-0.113

-0.936***

-0.321***

(0.460)

(0.328)

(0.286)

(0.183)

(0.053)

Sum of Transfers

0.035

0.035

0.076***

0.057***

0.022***

(0.045)

(0.050)

(0.024)

(0.018)

(0.007)

Equal Transfers

0.875

0.183

0.512*

0.487**

0.191**

(0.561)

f (0.363)

(0.281)

(0.199)

(0.078)

Final Payments are Closer

-0.179

0.357

0.468**

0.340**

0.129**

(0.411)

(0.297)

(0.219)

(0.157)

(0.058)

Game 1

0.608***

0.237***

(0.214)

(0.082)

Game 2

1 .008***

0.385***

(0.212)

(0.075)

Constant

-0.401

-0.535

-2.858***

-2.146***

(1.387)

(1.198)

(0.721)

(0.543)

Observations

169

197

266

632

632

Number of Groups

28

28

32

88

88

Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%

APPENDIX E

This appendix presents the two-player versions of two recent social-preference
models, and considers how such preferences affect the possibility and likelihood of
(mutual) cooperation in relation to characteristics of qualifying transfer pairs. In what
follows below, by more egalitarian transfers we mean those transfers that bring players'
material payoffs from mutual cooperation closer to each other. When transfers make the
material payoffs from mutual cooperation identical, we simply call them egalitarian
transfers.

Fehr and Schmidt (1999)

Denote by n. player fs material payoff. Fehr and Schmidt (1999) introduces the
following utility function (in the two-player case):

V j (jt i ,Jtj) = Jt i - a j max {k j - n i , 0} - /3, max {jt j - jtj , 0} , (1)
where /3 . < a t , < < 1 . We define a Social Welfare Equilibrum (SWE) of a game with
material payoffs as a Nash equilibrium of the game with material payoffs replaced with
payoff functions V i as in (1).

We demonstrate that for all three games, (i) unless the transfers are egalitarian, the player
with a smaller material payoff from (C, C) sometimes has incentives to deviate from C to
D; and (ii) mutual defection (D, D) is always a SWE in the second stage.

Game 1

In game 1 , egalitarian transfer pairs are characterized by the equality of y - x = 6 .
The defector's material payoff is always bigger than the cooperator's material payoffs
with any transfer pair in the SPE region. That is, 52 - x > 8 + x and 60 - y > 8 + y for all
transfer pairs (x, y) in the SPE region. Hence, by (1)

V 1 (D,C) = (52-x)-P 1 (44-2x). (2)
V 2 (C,D) = (60 -y)-p 2 (52-2y). (3)

Transfers with y - x > 6 .

In this case, player 1 's material payoff from (C, C) is no less than that of player 2. By
(1),

V X (C,C) = 40 + y - x - ^[2{y -x)- 12] = 40 + (1 - 2fa)(y -x) + 12/3,. (4)
V 2 (C,C) = 52 + x - y - a 2 [2(y -x)- 12] = 52 - (1 + 2a 2 )(y -x) + \2a v (5)

By (2) and (4),

V x (C, C) > V x (D, C) if and only if (y -12) + (56 - 2y)p 1 > 0.

Since 12 < y < 20 for any transfer pairs (x, y) in the SPE region, the above necessary and
sufficient condition always holds in the SPE region. On the other hand, by (3) and (5),

V 2 (C, C) > V 2 (C, D) if and only if (x - 8) + a 2 [12 + 2(x - y)] + fi 2 (52 - 2y) > 0.
Notice the second term on the left-hand- side of the second inequality is negative
when y - x > 6 . Hence, since y - x > 6 , whether the necessary and sufficient condition

holds depends on the sizes ofa 2 ,/3 2 . Player 2 may thus have incentives to deviate from C
to D unless the transfers are egalitarian, i.e. when y - x = 6.

Transfers with y - x < 6 .

In this case, player 1 's material payoff from (C, C) is no bigger than that of player 2. By
(1),

V l (C,C) = 40 + y - x - a,[12 + 2(x - y)] = 40 + (1 + 2a,)(y -x)- 12a, . (6)
V 2 (C, C) = 52 + x - y - [S 2 [12 + 2(x - y)] = 52 + (2/3 2 - l)(y - x) - U/3 2 . (7)
By (2) and (7),

V 2 (C, C) > V 2 (C, D) if and only if(x - 8) + /3 2 (40 - 2x) > 0.
Since 8 < x < 16 and 12 < y < 20 for any transfer pairs in the SPE region, it follows that the
above necessary and sufficient condition always holds for transfer pairs in the SPE
region. On the other hand, by (2) and (6),

V l (C, C) > V l (D, C) if and only if (y - 12) - a , [12 + 2(x - y)] + /3 l (44 - 2x) > 0.
The second term one the left-hand- side of the above inequality is negative
when y - x < 6. Since y - x s 6 , it follows that whether the necessary and sufficient

condition holds depends on the sizes ofa.\,fi A . Player 1 may therefore have incentives to
deviate from C to D unless transfers are egalitarian.

Game 2

In Game 2, egalitarian transfer pairs are characterized by the equality
of y - x = 10 . It turns out that no transfer pairs in the SPE region satisfies this condition.
The defector's material payoff is always bigger than the cooperator's material payoff with
any transfer pair in the SPE region. That is, 40 - x > 8 + x and 4 + y < 60 - y for any
transfer pair (x, y) in the SPE region. Hence, by (1),

F,(AC) = 40-x-i3,(32-2x). (8)
V 2 (C,D) = 60- y-p 2 (56-2y). (9)
Notice that in Game 2, y - x > 10 is not possible because y max - x min = 8 . Hence, we have
y - x < 10 for all transfer pairs (x, y) in the SPE region. That is, player 1 's material payoff
from (C, C) is always less than that of player 2 over the SPE region. By (1),

V x (C, C) = 32 + y - x - a, [(20 + 2(x - y)] = 32 + (1 + 2a, )(y - x) - 20a, . (1 0)
V 2 (C, C) = 52 + x - y - 13 2 [20 + 2(x - y)] = 52 + (2/3 2 - l)(y -x)- 20/3 2 . (1 1)
By (9) and (11),

V 2 (C, C) > V 2 (C, D) if and only if (x - 8) + /3 2 (36 - 2x) > 0.

Since 8 < x < 16 for any transfer pair (x, y) in the SPE region, the above necessary and
sufficient condition always holds. On the other hand, by (8) and (10),

V x (C, C) > V x (D, C) if and only if (y - 8) - a, [(20 + 2(x - y)] + ft (32 - 2x) > 0.
Since y - x < 10 , it follows that the second term on the left-hand- side of the second
inequality is always negative. Thus, for any a 1? ft satisfying the stated conditions, there
always exist transfer pairs within the SPE region that would make the second inequality
unsatisfied. Hence, for any a^ft satisfying the stated conditions, there ways exist
transfer pairs that would eliminate (C, C) as a SWE.

Game 3

In game 3, egalitarian transfers are characterized by the condition y - x = -4 .
However, in this game the defector's payoff is not always bigger than that of the
cooperator.

Transfers with y-x> -4.

In this case, player l's material payoff from (C,C) is no less than that of player 2. By (1),
V 1 (C,C) = 44 + y-x- ft [8 + 2(y - x)] = 44 + (1 - 2ft ){y - x) - 8ft. (12)
V 2 (C,C) = 36 + x - y - a 2 [S + 2(y - x)] = 36 - (1 + 2a 2 )(y - x) - 8a 2 . (13)

Case l:8<x<26, 8<j<18.

In this case, the defector's material payoff is no less than that of the cooperator at either
(D, C) or (C,D). Thus, by (1),

V 1 (D,C) = 52-x-p l (52-2x). (14)

V 2 (C,D) = 44-y-l3 2 (36-2y). (15)

By (12) and (14),

V x (C, C) > V x (D, C) if and only if (y-S) + ft (44 - 2y) > 0. (1 6)
Since 8 < y < 18 , the necessary and sufficient condition clearly holds. Hence,
V X (C,C) > V X (D,C) . By (13) and (15),

V 2 (C, C) > V 2 (C, D)) if and only if (x - 8) - a 2 [8 + 2(y - x)] + ft (36 - 2y) > 0.
In the range of 8 < x < 26, 8 < y < 18 , and y - x > -4 , there are transfer pairs for given
a 2 ,ft such that the above necessary and sufficient condition does not satisfy. With these
transfer pairs, (C, C) will be eliminated as a SWE.

Case 2:26<x<28,8<>-<18.

In this case, y - x < -8 . Thus, the range with 26 < x < 28, 8 < y <, 1 8, and y - x > -4 is
empty.

Case 3: 8 < x < 26, 18 < y <. 24.

In this case, the defector's material payoff at (D, C) is larger than that of the cooperator
wile the opposite holds at (C, D). Thus, V { (D, C) is as in (14). By (16),
V l (C, C) > V l (D, C) if and only if (y - 8) + ft (44 - 2y) > 0. Since ft < 1 , the preceding
necessary and sufficient condition clearly holds. On the other hand, by (1)

V 2 (C,D) = 44- y-a 2 (2y -36). (17)

Thus, by (13) and (17),

V 2 (C,C) > V 2 (C,D) if and only if (x - 8) + a 2 (2x - 44) > 0. (18)
Since 8 < x < 26 , the above necessary and sufficient condition does not always hold.
Thus, there are transfer pairs with which (C, C) is eliminated as a SWE.

Case 4: 26 < x < 28, 18 < y < 24 .

In this case, the defector's material payoff is always less than that of the cooperator' s. It
follows that

V i (D,C) = 52-x-a 1 (2x-52) (19)
while V 2 (C,D) is as in (17). By (12) and (19),

V l (C, C) > V i (D, C) if and only if(y-S) + a l (2x - 52) - ft [8 + 2(y - x)] > 0.
Notice in this case, y > 22 in order for y - x > -4 . Consequently, 8 + 2(y - x) < 4 which
implies - ft [8 + 2(y - x)] > -4ft . Since ft < 1 and y - 8 > 14 , the above necessary and
sufficient condition is satisfied. On the other hand, by (18), V 2 (C,C) > V 2 (C,D) if and
only if (x - 8) + a 2 (2x - 44) > . This condition is clearly satisfied in the range of
26 <. x < 28 .

A parallel analysis can be established for Transfers with y - x < -4 in the SPE region of
Game 3. It can also be verified that for all three games, (D, D) will always be a SWE with
transfers in the SPE regions.

An example

Consider Game 2:

Player 1

Game 2

Player

r ?

C

D

c

32, 52

4, 60

D

40, 8

20, 24

Suppose (H,, H 2 ) = (15, 9). Then the transformed game is:

Game 2 transformed by (15, 9)

Player 2

Player 1

C

D

c

26,58

13,51

D

25,23

20, 24

Suppose the players have Fehr-Schmidt preferences. The utility from each outcome is:

Game 2 transformed by (15, 9), F-S utility

Player 2

Player 1

C

D

c

26-32a, 58-32(3

13-38a, 51-38(3

D

25-2(3, 23-2a

20-4a, 24-4(3

If Player 2 chooses C, Player 1 's best response depends on the values of a and (3. If 25-
2(3 > 26-32oc, or 32oc-2|3 > 1, then D is the best response. Since (3 cannot exceed a, if
30a > 1, (C,C) is not an equilibrium.

If Player 1 chooses C, then if 51-38(3 > 58-32(3, or -6(3 > 7, then D is Player 2's best
response. But since (3 > 0, this cannot occur, so C is Player 2's best response to C by
Player 1.

Now suppose instead that (H^ H 2 ) = (9, 15) and that players have F-S preferences. The
utility from each outcome is:

Game 2 transformed by (9, 15), F-S utility

Player 2

Player 1

C

D

c

38-8oc, 46-8(3

19-26a, 45-26(3

D

31-14(3, 17-14a

20-4a, 24-4(3

If Player 2 chooses C, Player 1 's best response depends on the values of a and |3. If 3 1-
14(3 > 38-8a, or 8a- 14(3 > 7, then D is the best response. Even if (3 = (the minimum
value), we must have 8a > 7 for D to be a best response, so that a must be at least 7/8.

If Player 1 chooses C, then if 45-26(3 > 46-8(3, or -18(3 > 1, then D is the best response.
But since (3 > 0, this cannot occur, so C is Player 2's best response to C by Player 1.

Overall, in this example, the more egalitarian transfers make cooperation for Player 1 a
best response for a broader range of values, while not affecting the range for Player 2

Thus, (C,C) is an equilibrium for a broader range of values when transfers bring the
mutual-cooperation payoffs closer together than further apart.

Charness and Rabin (2002)

Charness and Rabin (2002) introduces a(A,<5) - utility function for each player:
V^ji^jIj) = Ji i + A(l - d)jij,if 7t i < Jij.; V^jt^jTj) = (1 - kd)n i + kjt . if Jt i > jt ., (1)
where A, <5 G[0, 1] and n i and;r y are the material payoffs of payers i and j. Note that this

is the reciprocity-free version of the full model. They define a Social Welfare Equilibrium
(SWE) of a game with material payoffs as a Nash equilibrium of the game with material
payoffs replaced with (A,<5) - social welfare payoffs.

We now demonstrate the following properties for the three games in the paper.
(A) Mutual cooperation is more socially rewarding (both players benefit) the more
egalitarian the transfers are when A(l + <5) > 1 ; the player receiving the lower material
payoff from mutual cooperation prefers more egalitarian transfers to less egalitarian
transfers while the other player holds opposite preferences when A(l + <5) < 1 . (B)
Mutual cooperation is always a SWE in the second stage for all transfer pairs within the
SPE regions. (C) Mutual defection is eliminated as a SWE in the second stage for a range
of transfer pairs in the SPE regions of Game 1 and Game 2, but mutual defection is
always a SWE in the second stage for all transfers in the SPE region of Game 3.

Game 1

In game 1 , egalitarian transfer pairs are characterized by the equality of y - x = 6 .
Furthermore, the defector's material payoff is always bigger than the cooperator's
material payoffs with any transfer pair in the SPE region. That is,
52 - x > 8 + x and 60 - y > 8 + y for all transfer pairs (x, y) in the SPE region. By (1)

V x (D,C) = (1 - A<5)(52 -x) + A(8 + x) = [A(l + 8) - l]x + 52(1 - A<5) + 8A. (2)
V 2 (C,D) = (1 - A<5)(60 - y) + A(8 + y) = [A(l + 8) - l]y + 60(1 - A<5) + 8A. (3)

Transfers with y - x ^ 6 .

In this case, player 1 's material payoff from (C, C) is no less than that of player 2. By (1),
V x (C,C) = [1 - A(l + 8)](y - x) + 44(1 - A<5) + 52A. (4)

V 2 (C,C) = -[1 - A(l - 8 )]( j - x) + 44A(1 -<5) + 52. (5)
With y - x > 6 , (4) and (5) imply that as transfers become more egalitarian, both players'
C-R payoff functions increase in the difference y - x when A(l + 8) > 1 ;
V x (C, C) decreases while V 2 (C, C) increases in y - x when A(l + 8 ) < 1 . This shows that
mutual cooperation is more socially rewarding the more egalitarian the transfers are when
A(l + 8) > 1 ; the player receiving the lower material payoff from mutual cooperation
prefers more egalitarian transfers to less egalitarian transfers while the other player holds
opposite preferences when A(l + 8) < 1.

By (2) and (4),

V x (C, C) > V x (A C) if and only if 36A > [A(l + <5) - \](y - 8).
Since [A(l + 8) - \](y - 8) < A<5(j - 8) and 12 < y < 20 for any transfer pairs (x, y) in the
SPE region, [A(+<5) - \](y - 8) < 12A<5 . It follows that the above necessary and sufficient
condition always holds in the SPE region for any A, 8 E[0, 1] .

Similarly, by (3) and (5),

V 2 (C,C) > V 2 (C,D) if and only if (x - 8) + A(36 - x) > A<5(j - 8) + k8(y - x - 8).
Since 8 < x < 16 and 12 < y < 20 for any transfer pairs (x, y) in the SPE region, it follows
that y - 8 < 12and y - x - 8 < 4 . Consequently, A<5(j - 8) + A<5(j - x - 8) < 16A<5 . Since
8 < 1 , the above necessary and sufficient condition always holds in the SPE region. This
shows that (C, C) is always a SWE. A parallel analysis can be made for transfers
satisfying y - x < 6 .

Game 2

In Game 2, egalitarian transfer pairs are characterized by the equality
of y - x = 10 . It turns out no transfer pairs in the SPE region satisfies this condition.
Furthermore, the defector's payoff is always bigger than the cooperator's material payoff
with any transfer pair in the SPE region. That is, 40 - x > 8 + x and 4 + y < 60 - y for any
transfer pair in the SPE region. Hence, by (1),

V X {D,C) = [A(l + 8)- l]x + 40(1 - A<5 ) + 8 A. (8)
V 2 (C,D) = [A(l + 8)- l]y + 60(1 - A5 ) + 4A. (9)
y - x > 10 is not possible because y max - x min = 8 .

Transfers with y - x < 1 .

In this case, player 1 's material payoff is less than that of player 2. By (1),
V x {C,C) = [1 - A(l - 8)](y -x) + 52A(1 - <5) + 32. (10)
V 2 {C,C) = -[1-A(1 + 8 )](y-x) + 52(1- A<5) + 32A. (11)

By (10) and (11), y - x < 10 implies that as transfers become more egalitarian, both
players' C-R payoff functions increase in the difference y-x when A(l + 8) > 1 ;

V x (C, C) increases while V 2 (C, C) decreases in y - x when A(l + <5 ) < 1 . This shows that
mutual cooperation is more socially rewarding the more egalitarian the transfers are
when A(l + <5) > 1 ; the player receiving the lower material payoff from mutual
cooperation prefers more egalitarian transfers to less egalitarian transfers while the more
materially paid player hold opposite preferences when A(l + 8) < 1 .

By (8) and (10),

V X (C,C) > V X (D,C) if and only if A(16 - ox) + A[(16 - y) + 8{y - x)] > 0.
Since 8 < x < 16 and 8 < y < 16 for any transfer pair (x, y) in the SPE region, the
necessary and sufficient condition always holds. Similarly, by (9) and (11),
V 2 (C, C) > V 2 (C, D) if and only if 28A > Ax + (A<5 - \)(x - 8).
Since 8 < x < 16 andA<5 < lfor any transfer pairs (x, y) in the SPE region, the necessary
and sufficient condition always holds. This shows that (C, C) is always a SWE.

Game 3

In Game 3, egalitarian transfers are characterized by the condition y - x = -4 .
However, in this game whether the defector's payoff is bigger than that of the cooperator
depends on the transfers.

Transfers with y - x > -4 .

In this case, player l's material payoff from (C,C) is no less than that of player 2. By (1),
V l (C, C) = [1 - A(l + 8)](y -x) + 44(1 - A<5) + 36A. (12)
V 2 {C,C) = -[1-A(1- 3 )](j - x) + 44A(1 -<5) + 36. (13)
From (12) and (13), y - x > -4 implies that as transfers become more egalitarian, Both
players' C-R payoff functions increase in y - x whenA(l + 6) > 1 ; V X {C,C) decreases
while V 2 (C, C) increases my - x when A(l + 8) < 1. This shows that mutual cooperation
is more socially rewarding the more egalitarian the transfers are when A(l + <5) > 1 ; the
less materially paid player from mutual cooperation prefers more egalitarian transfers to
less egalitarian transfers while the more materially paid player hold opposite preferences
when A(l + 8) < 1.

Case l:8<x<26, 8<j<18.

In this case, the defector's payoff is no less than that of the cooperator at either (D, C) or
(C,D).Thus,by(l),

V x (D,C) = [A(l + 8) - l]x + 52(1 - A<5). (14)
V 2 (C,D) = [A(l + 8) - l]y + 44(1 - A<5) + 8A. (15)

By (12) and (14),

V x (C, C) > V x (D, C) if and only if 36A > ky + (A<5 - \)(y - 8).

Since 8 < y < 18 and since A<5 < 1 , the necessary and sufficient condition clearly holds.
Hence, V X (C,C) > V X (D,C). By (13) and (15),

V 2 (C,C) > V 2 (C,D)) if and only if [1 - A(l - 8)](x - 8) + A(36 - 28y) + 28A(1 - 3) > 0.
Since 8 < x < 26, 8 < _y < 18 and since A, <5 G[0,1] , the necessary and sufficient condition
holds. Hence, V 2 (C,C) > V 2 (C,D) .

Case 2:26<x<28,8< j/<18.

In this case, y - x < -8 .The range with 26 < x < 28, 8 < y < 1 8, and y - x > -4 is thus
empty.

Case 3: 8 < x < 26, 18 < y < 24.

In this case, the defector's material payoff at (D, C) is larger than that of the cooperator
while the opposite holds at (C, D). Thus, V 1 (D, C) is as in (14).
Hence, F,(C,C) > V l (D,C)as shown in case 1. By (1),

V 2 (C,D) = [A(l - 8) - l]y + 8A(1 - <5) + 44. (16)
Thus, by (13) and (16),

V 2 (C,C) > V 2 (C,D) if and only if (x - 8) + A(l - <5)(36 - x) > 0.
Since 8 < x < 26 , the second inequality holds. Hence, V 2 (C, C) > V 2 (C, D) .

Case 4: 26 < x < 28, 18 < y < 24 .

In this case, the defector's material payoff is always less than that of the cooperator' s. It
follows that

V l (D,C) = 52-x + ?i(l-8)x. (17)

By (12) and (17),

V x (C, C) > V x (D, C) if and only if A<5 (x - y) + A<5 (x - 44) + (y - 8) + (36 - y)k > 0.
Since 26 < x < 28, 1 8 < y < 24 , and y - x > -4 , we have y > 22 . Hence,

A<5(x - j) + A<5(x - 44) + j; - 8 + (36 - j)A > 2A<5 - 18A<5 + 14 + 12A > 0.
Consequently, V x (C, C) > V x (D, C) . On the other hand, V 2 (C, D) is as in ( 1 6). Hence,
V 2 (C,C) > V 2 (C,D) as shown in Case 3.

In summary, we have shown that V x (C, C) > K t (Z), C) and F 2 (C, C) > V 2 (C, D) for all
transfer pairs in the SPE region with y - x > -4 . Hence, (C, C) is always a SWE for
transfer pairs in the SPE region satisfying y - x > -4 . A parallel analysis can be made for
transfer pairs in the SPE region satisfying y - x < -4 .

Elimination of (D, D) as a Social Welfare Equilibrium

We show that mutual defection does not survive social considerations as modeled in
Charness and Rabin (2002) over a large range of transfer pairs in the SPE regions in
Games 1 and 2; it survives in Game 3 over the entire region of transfer pairs.

Game 1

Notice in this game

V 1 (C,D) = 8 + y + ?i(l-d)(60-y) (18)
F 1 (£>,£>) = 28(1-A<5) + 24A. (19)

By (18) and (19),

V X (C,D) > V,(D,D) if and only if y > 16 - ^f^r -

1 - A(l - O )

Since (x, y) is in the SPE region if and only 8 < x < 16 and 12 < y < 20 , there exist many
transfer pairs that make V { (C, D) > V l (D, D). With these transfer pairs, (D, D) cannot be
aSWE.

Game 2

In this game,

V x (C, D) = 4 + y + A(l - 8 )(60 - y). (20)
F,(A£>) = 20 + 24A(l-(5). (21)
Together, (20) and (21) imply

V x (C, D) > V x (D, D) if and only if y> 16 -

1 - A(l - O)

Since (x, y) is in the SPE region if and only if 8 < x < 16 and 8 < y < 16 , there exist many
transfer pairs that make V { (C, D) > V l (D, D). With these transfer pairs, (D, D) cannot be
aSWE.

Game 3

Notice first in this game

F,(A£>) = 32(1-A<5) + 28A. (22)

F 2 (A£>) = 28 + 32A(1-(S). (23)
We partition the SPE region into four different parts, depending on the comparison of a
defector's material payoff with that of the defector.

Case l:8<x<26, 8<j<18.
In this case,

V x (C, D) = 8 + y + A(l - <5)(44 - y). (24)
V 2 (D,C) = x + M\-8)(52-x). (25)

By (22) and (24),

V X (C,D) < V X (D,D) if and only if A(l - <5)(12 - y) < 24 - y - 4k.
Since 24 - y - 4A < 20 - y and since A(l - <5) < 1 , the necessary and sufficient condition
holds. By (23) and (25),

V 2 (D, C)) < V 2 (D, D) if and only if A(l - <5 )(20 - x) < 28 - x.
Since A(l - <5) < 1 , the necessary and sufficient condition holds.

Case 2:26<x<28,8< y<\8.

In this case, V X {C,D) is as in (24). Hence, V X {C,D) < V^(D,D) as shown in Case 1. For
player 2,

F 2 (AC) = (l-A<5)x + A(52-x). (26)
Together, (23) and (26) imply

V 2 (D,Q) < V 2 (D,D) if and only if A(52 - 2x) < [1 + A(l - <5)](28 - x) + 4A(1 - <5).
Since 26 < x < 28 , the necessary and sufficient condition holds.

Case 3: 8 < x < 26, 18 < y < 24.

In this case, V 2 (D,C) is as in (25). Hence, V 2 (D, C) < V 2 (D,D) as shown in Case 1. For
player 1,

V 1 (C,D) = (l-X8)(S + y) + k(44-y). (27)

By (22) and (27),

V X {C,D) < V X {D,D) if and only if A(16 - y) < (1 - A5)(24 - y).
Since 18 < y < 24 , the necessary and sufficient condition holds.

Case 4: 26 < x < 28, 18 < y < 24 .

In this case, V x (C, D) is as in (27) and V 2 (D, C) is as in (26).

Hence, V x (C, D) < V x (D, D) as shown in Case 3 and V 2 (D, C) ^ V 2 (D, D) as shown in Case
2.

In summary, we have shown that (D, D) is always a SWE for all transfer pairs in the SPE
region of Game 3.

An example

Consider Game 2:

Game 2

Player 2

C

D

Player 1

c

32, 52

4, 60

D

40, 8

20, 24

Suppose (H l5 H 2 ) = (15, 9). Then the transformed game is:

Game 2 transformed by (15, 9)

Player 2

Player 1

C

D

c

26,58

13,51

D

25,23

20, 24

Suppose the players have Charness -Rabin distributional preferences. The utility from
each outcome is:

Game 2 transformed by (15, 9), C-R utility

Player 2

Player 1

C

D

c

26+58X(l-6), 58(1- X6)+ 26X

13+51X<l-6), 51(1- X6)+ 13X

D

25(1- X6)+ 23X, 23+25X(l-6)

20+24X0-6), 24(1- X6)+ 20X

If Player 2 chooses C and if 25(1- X6) + 23X > 26 + 58X(l-6), or 25 - 25X6 + 23X > 26 +
58X- 58X6, or 336X - 35X > 1, then Player 1 's best response is D. But since 6 cannot
exceed 1, this condition cannot hold, so C is always Player 1 's best response to C from
Player 2.

If Player 1 chooses C, Player 2's best response is D if 51(1- X6)+ 13X > 58(1- X6)+ 26X,
or 51 - 51X6 + 13X > 58 - 58X6 + 26X, or 76X - 13X > 7. But since 6 and X cannot exceed
1, 75X cannot exceed 7, and since X is non-negative, 75X - 13X cannot exceed 7. Thus,
this condition can't hold, so that C is always Player 2's best response to C from Player 1.

Now suppose instead that (H,, H 2 ) = (9, 15) and that players have C-R preferences. The
utility from each outcome is:

Game 2 transformed by (9, 15), C-R utility

Player 2

Player 1

C

D

c

38+46X(l-6), 46(1-X6)+38X

19+45X(l-6), 45(1- X6)+ 19X

D

31(1- X6)+ 17X, 17+31^(1-6)

20+24X(l-6), 24(1- X6)+ 20X

If Player 2 chooses C and if 31(1- X6) + 17X > 38 + 46X(l-6), or X(156 - 29) > 7, then
Player 1 's best response is D. But since 6 cannot exceed 1, this condition cannot hold, so
C is always Player 1 's best response to C from Player 2.

If Player 1 chooses C and if 45(1- XS) + 19X > 46(1- X6) + 38X, or > (1 - XS) + 19X,
then Player l's best response is D. But since 5 and X cannot exceed 1,(1- XS) cannot be
negative, and since X is non-negative, (1 - X6) + 19X must be positive. Thus, this
condition can't hold, so that C is always Player 2's best response to C from Player 1.

In these examples, (C,C) is an equilibrium for all permitted values of X and 5 when the
transfers are in the qualifying range.