The conventional wisdom on altruism in 2 x 2 games, as I take it: Although altruism can contribute to higher social value, the altruist is destined to lose out to the egoist in their interactions. Hence altruism will decline over time, absent sufficient ability of altruists to form their own groups and/or to punish those who do not behave altruistically. The modification to conventional wisdom argued for here: Altruists in games in which mixed Nash strategies are an equilibrum in some cases do better in those equilibria than egoists. More specifically, altruists do better than egoists in their interactions when the altruist's known altruism leads the egoist to play the altruist's objectively preferred strategy more often than would be the case absent the egoist's knowing he/she is playing with an altruistic player. Yet more specifically: If the effect of altruism is to narrow the difference in the altruist's payoffs for the egoist strategy that is *more* preferred by the altruist relative to the difference in payoffs for the egoist's strategy that is *less* preferred by the altruist, altruism results in the altruist outperforming the egoist. In the reverse case in which altruism widens the difference in the altruist's payoffs for the strategy he/she prefers the egoist play relative to the less-preferred strategy, altruism is harmful to the altruist. The takeaway: Although the cases are symmetrical overall, with the losses from altruism balancing out the gains, there is an important class of 2 x 2 games that broadly relate to leading and following (Chicken, Leadership, and Battle of the Sexes) in which altruism helps the altruist to induce the rational egoist to defer to the altruist. A concern on the part of leaders and managers for the welfare of those they lead and manage can thus be understood as strategically efficacious.
What follows are matrices for the twelve basic types of symmetrical 2 x 2 games, all played between an egoist and a Benthamite altruist who values the welfare of the other equally with his/her own, that illustrate the foregoing points, along with a concluding summary of the gains and losses from altruism in the different games. (As will be noted, some of 3-2-1-0 payoffs are modified so that no dominant strategies apply for either player, which allows mixed Nash to be a plausible solution to the games.)
GAME 1: “LIFE IS GREAT”
a) Life is Great with Egoists
|
Life is Great--
ROW payoffs in lower left—COL
payoffs in upper right..
|
COL--
Cooperate
|
COL--
Defect
|
|
ROW--
Cooperate
|
3
3
|
1
2
|
|
ROW--
Defect
|
2
1
|
0
0
|
Egoistic equilibrium: R and C
cooperate--(3, 3)
b) Life is Great--ALT-EGO:
|
Life is Great--
ALT payoffs in lower left—EGO
payoffs in upper right..
|
EGO--
Cooperate
|
EGO--
Defect
|
|
ALT--
Cooperate
|
3
6
|
2
3
|
|
Row--
Defect
|
1
3
|
0
0
|
ALT-EGO equilibrium: ALT and EGO
cooperate--(3, 3)
GAME 2: “LIFE IS GOOD”
d) Life is Good for Egoists:
|
Life is Good--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Cooperate
|
Column--
Defect
|
|
Row--
Cooperate
|
3
3
|
0
2
|
|
Row--
Defect
|
2
0
|
1
1
|
Egoistic equilibrium: R and C
cooperate--(3, 3)
e) Life is Good for ALT-EGO:
|
Life is Good--
ALT payoffs in lower left—EGO
payoffs in upper right..
|
EGO--
Cooperate
|
EGO--
Defect
|
|
ALT--
Cooperate
|
3
6
|
0
2
|
|
ALT--defect
|
2
2
|
1
2
|
ALT-EGO mixed Nash equilibrium: Both
cooperate--(3, 3)
GAME 3: “RATIONAL TRUST”
g) Rational Trust for egoists:
|
Rational
Trust--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Trust
|
Column--
Distrust
|
|
Row--
Trust
|
3
3
|
2
1
|
|
Row--
Distrust
|
1
2
|
0
0
|
Egoistic equilibrium: R and C
cooperate--(3, 3)
h) Rational Trust for
ALT-EGO:
|
Rational
Trust--
Row payoffs in lower left—Column
payoffs in upper right..
|
EGO--
Cooperate
|
EGO--
Defect
|
|
ALT--
Cooperate
|
3
6
|
2
3
|
|
ALT--
Defect
|
1
3
|
0
0
|
ALT-EGO equilibrium: Both
cooperate--(3, 3)
GAME 4: “TRUST”
j)
Trust for egoists:
|
Trust--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Trust
|
Column--
Distrust
|
|
Row--
Trust
|
3
3
|
2
0
|
|
Row--
Distrust
|
0
2
|
1
1
|
Egoistic mixed Nash equilibrium: .5
trust--(1.5, 1.5)
k) Trust for ALT-EGO (with payoffs adjusted so trust is not dominant for ALT):
|
Trust--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Trust
|
Column--
Distrust
|
|
Row--
Trust
|
3
6
|
2
2
|
|
Row--
Distrust
|
0
2
|
1.5
3
|
ALT-EGO equilibrium: ALT trust 3/5, EGO trust 1/7--(5/7 ALT, 57/35 EGO)
GAME 5: “SCHELLING”
m) Schelling for egoists:
|
Schelling--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Focal
|
Column--
Non-Focal
|
|
Row--
Focal
|
3
3
|
0
1
|
|
Row--
Non-Focal
|
1
0
|
2
2
|
Egoistic mixed Nash equilibrium: .25
focal--(1.5, 1.5)
n) Schelling for ALT-EGO:
|
Schelling--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Focal
|
Column--
Non-Focal
|
|
Row--
Focal
|
3
6
|
0
1
|
|
Row--
Non-Focal
|
1
1
|
2
4
|
ALT-EGO mixed Nash equilibrium: ALT .25 focal, EGO 3/8 focal--(ALT 1 3/8, EGO 1.5)
GAME 6: “STAG HUNT”
p) Stag Hunt for egoists:
|
Stag Hunt--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Stag
|
Column--
Hare
|
|
Row--
Stag
|
3
3
|
1
0
|
|
Row--
Hare
|
0
1
|
2
2
|
Egoistic mixed Nash equilibrium: .5
stag--(1.5, 1.5)
q) Stag Hunt for ALT-EGO:
|
Stag Hunt--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Stag
|
Column--
Hare
|
|
Row--
Stag
|
3
6
|
1
1
|
|
Row--
Hare
|
0
1
|
2
4
|
ALT-EGO mixed Nash equilibrium: ALT .5 stag, EGO 3/8
stag--(ALT 1 3/8, EGO 1.5)
GAME 7: “CHICKEN”
s) Chicken for egoists:
|
Chicken--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defer/Cooperate
|
Column--
Assert/Defect
|
|
Row--
Defer/Cooperate
|
2
2
|
3
1
|
|
Row--
Assert/Defect
|
1
3
|
0
0
|
Egoistic mixed Nash equilibrium: .5
defer/cooperate--(1.5, 1.5)
t) Chicken for ALT-EGO (payoffs modified so COOP is not dom for ALT):
|
Chicken--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defer/Cooperate
|
Column--
Assert/Defect
|
|
Row--
Defer/Cooperate
|
2
4
|
3.5
5
|
|
Row--
Assert/Defect
|
1.5
4.5
|
0
0
|
ALT-EGO mixed Nash equilibrium: ALT assert 1/2, EGO assert 1/11--(ALT 113/44, EGO 77/44)
GAME 8: “PRISONER’S DILEMMA”
v) Prisoner’s Dilemma for
egoists:
|
Prisoner’s Dilemma--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Cooperate
|
Column--
Defect
|
|
Row--
Cooperate
|
2
2
|
3
0
|
|
Row--
Defect
|
0
3
|
1
1
|
Egoistic equilibrium: Defect--(1, 1)
w) Prisoner’s Dilemma for
ALT-EGO:
|
Prisoner’s Dilemma--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Cooperate
|
Column--
Defect
|
|
Row--
Cooperate
|
2
4
|
3
3
|
|
Row--
Defect
|
0
3
|
1
2
|
ALT-EGO equilibrium: ALT Coop EGO defect--(ALT 0, EGO 3)
GAME 9: “LEADERSHIP”
y) Leadership for egoists:
|
Leadership--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Follow
|
Column--
Lead
|
|
Row--
Follow
|
1
1
|
3
2
|
|
Row--
Lead
|
2
3
|
0
0
|
Egoistic mixed Nash equilibrium:
Follow .5--(1.5, 1.5)
z) Leadership for ALT-EGO:
|
Leadership--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Follow
|
Column--
Lead
|
|
Row--
Follow
|
1
2
|
3
5
|
|
Row--
Lead
|
2
5
|
0
0
|
ALT-EGO mixed Nash equilibrium:
ALT follow.75, EGO follow 5/8--(ALT 1 5/8, EGO 1.5)
GAME 10: “BATTLE OF THE SEXES”
bb) Battle of the Sexes for
egoists:
|
Battle of the Sexes --
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defer
|
Column--
Assert
|
|
Row--
Defer
|
0
0
|
3
2
|
|
Row--
Assert
|
2
3
|
1
1
|
Egoistic mixed Nash equilibrium: Defer
.25--(1.5, 1.5)
cc) Battle of the Sexes for
ALT-EGO:
|
Battle of the Sexes--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defer
|
Column--
Assert
|
|
Row--
Defer
|
0
0
|
3
5
|
|
Row--
Assert
|
2
5
|
1
2
|
ALT-EGO mixed Nash equilibrium:
ALT Defer .25, EGO Defer 3/8--(ALT 1 5/8, EGO 1.5)
GAME 11: “COVENTRY”
ee) Coventry for egoists:
|
Coventry--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defect
|
Column--
Cooperate
|
|
Row--
Defect
|
1
1
|
3
0
|
|
Row--
Cooperate
|
0
3
|
2
2
|
Egoistic equilibrium: Cooperate--(2,
2)
ff) Coventry for ALT-EGO:
|
Coventry--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defect
|
Column--
Cooperate
|
|
Row--
Defect
|
1
2
|
3
3
|
|
Row--
Cooperate
|
0
3
|
2
4
|
ALT-EGO equilibrium: Cooperate--(2, 2)
GAME 12: “LEVIATHAN”
hh) Leviathan for egoists:
|
Leviathan--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defect
|
Column--
Cooperate
|
|
Row--
Defect
|
0
0
|
3
1
|
|
Row--
Cooperate
|
1
3
|
2
2
|
Egoistic equilibrium: Cooperate--(2,
2)
ii) Leviathan for ALT-EGO:
|
Leviathan--
Row payoffs in lower left—Column
payoffs in upper right..
|
Column--
Defect
|
Column--
Cooperate
|
|
Row--
Defect
|
0
0
|
3
4
|
|
Row--
Cooperate
|
1
4
|
4
4
|
ALT-EGO equilibrium: COOP (2, 2)
ALT and EGO payoffs in the games ALT EGO
+ = mixed Nash games
1. Life is Great 3 3
2. Life is Good 3 3
3. Rational Trust 3 3
4. Trust* + 5/7 57/35
5. Schelling +
1 3/8 1.5
6. Stag Hunt + 1 3/8 1.5
7. Chicken + 113/44 77/44
8. Prisoner's Dilemma 0 3
9. Leadership + 1
5/8 1.5
10. Battle of the Sexes + 1
5/8 1.5
11. Coventry 2 2
12. Leviathan 2 2
Summary--ALT relative to EGO in MIXED NASH games--The advantage for ALT in Leadership and BOTS is exactly balanced by the advantage for EGO in Schelling and Stag Hunt. With the payoffs assigned here, ALT's advantage in Chicken (9/11) outweighs EGO's advantage in Trust (22/35)--given that this depends on specific numbers assigned, though, it is better to treat the two games as having matching and equal advantages for ALT and EGO pending a general, non-plug-in analysis. Note that the advantages of ALT are in the three games in which the optimal solution involves the players playing different strategies and receiving different payoffs.