UNSW | BUSINESS SCHOOL | SCHOOL OF ECONOMICS
Calling the shots
Experimental evidence on significant aversion
to non-existing strategic risk
Ben Greiner
Strategic Risk
 “Regular” risk:
• Uncertainty about an outcome of a random event
• Known or unknown (guessed, subjective) probabilities
 Strategic risk:
• Uncertainty about strategies/actions other people are
going to choose
• Subjective probabilities
 In (standard) game theory there is no strategic uncertainty
• Beliefs are correct in equilibrium, and equilibrium
strategies are best responses to beliefs
An experiment
A: $??
B: $??
B
A: $??
B: $??
A
A: $??
B: $??
B
A: $??
B: $??
A: $80
B: $20
B
A: $0
B: $0
A
A: $50
B: $50
B
A: $0
B: $0
A: $80
B: $20
B
A: $0
B: $0
A
A: $50
B: $50
B
A: $50
B: $0
 Ultimatum Game
 Reverse Impunity Game
A: 80
B: 20
B
A
B
A: 0
B: 0
A: 50
B: 50
A: 0
B: 0
A: 80
B: 20
B
A
B
A: 0
B: 0
A: 50
B: 50
A: 50
B: 0
Experiment 1
 Ultimatum Game
 Reverse Impunity Game
A: € 6.40
B: € 1.60
B
A
B
A: € 0.00
B: € 0.00
A: € 4.00
B: € 4.00
A: € 4.00
B: € 4.00
A: € 6.40
B: € 1.60
B
A
B
A: € 0.00
B: € 0.00
A: € 4.00
B: € 4.00
A: € 4.00
B: € 0.00
Experiment 1: Procedures
 One-shot experiment, pie size € 8, 145 participants
 Conducted in foyer of student restaurant:
• Students participant before lunch
• Get paid after lunch
 Participants decided in all (altogether 5) games and
both roles (strategy method)
 Payoff: 10-sided dice, thrown for 2 randomly
matched subjects, each number implies a particular
game and role assignment, game played out and
paid out
Experiment 1
 Ultimatum Game
91.0%
55%
B
A
B
 Reverse Impunity Game
93.8%
A: € 6.40
B: € 1.60
A: € 0.00
B: € 0.00
99.3% A: € 4.00
B: € 4.00
A: € 4.00
B: € 4.00
28%
B
A
B
A: € 6.40
B: € 1.60
A: € 0.00
B: € 0.00
99.3% A: € 4.00
B: € 4.00
A: € 4.00
B: € 0.00
Experiment 2: Beliefs
 2 years after first experiment in large classroom
 135 subjects were asked to predict frequencies of
choices in 1st experiment
 Instructions included all procedural information and
original instructions of 1st experiment
 Incentive compatible payment (step-function
approximately following quadratic scoring rule)
Experiment 2: Beliefs
Guessed Acceptance Rates of Equal Split Proposals
 Ultimatum Game  Reverse Impunity Game
~
99.3%
98.8%
100%
Exp 1 observed
Exp 2 Average Guess
Exp 2 Median Guess
99.3%
96.6%
100%
≥
100
100
80
80
60
60
40
40
20
20
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Experiment 2: Beliefs
Guessed Share of Unequal Split Proposals
 Ultimatum Game  Reverse Impunity Game
>
>
55.2%
79.0%
89.5%
Exp 1 observed
Exp 2 Average Guess
Exp 2 Median Guess
28.3%
51.6%
48.1%
100
100
80
80
60
60
40
40
20
20
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Experiment 3
 Ultimatum Game
 Reverse Impunity Game
A: € 12.0
B: € 3.0
B
A
B
A: € 0.0
B: € 0.0
A: € 7.5
B: € 7.5
A: € 0.0
B: € 0.0
A: € 12.0
B: € 3.0
B
A
B
A: € 0.0
B: € 0.0
A: € 7.5
B: € 7.5
A: € 7.5
B: € 0.0
Experiment 3: Play method
 214 participants invited to laboratory
 Role lottery  Responders leave lab to next room
 56 / 51 proposer-responder pairs in UG / RIG
 Pen & paper
Experiment 3
 Ultimatum Game
55%
64%
70%
B
A
A: € 12.0
B: € 3.0
A: € 0.0
B: € 0.0
94%
B
 Reverse Impunity Game
A: € 7.5
B: € 7.5
A: € 0.0
B: € 0.0
28%
39%
87%
B
A
A: € 0.0
B: € 0.0
100%
B
A: € 12.0
B: € 3.0
A: € 7.5
B: € 7.5
A: € 7.5
B: € 0.0
Experiment 4: Enter or not
 Ultimatum Game
 Reverse Impunity Game
 ENTER THE GAME or STAY OUT?
A: $ 31
B: $ 15
B
A
B
A: $ 10
B: $ 10
A: $ 23
B: $ 23
A: $ 0
B: $ 0
A: $ 31
B: $ 15
B
A
B
A: $ 10
B: $ 10
A: $ 23
B: $ 23
A: $ 23
B: $ 23
 Both ENTER: roles assigned 50/50; Both OUT: $20, $20
 One enters, other stays out: ENTER= proposer, OUT= responder
Experiment 4: Procedures
 66 / 52 participants in Ultimatum / Reverse Impunity
 Computerized, all decisions at computer screen, zTree
 Pairs of 2 participants anonymously
• decided simultaneously about entering
• dropped out or not
• were informed about random move, if necessary
• proceeded and played Ultimatum / Rev. Impunity
• got paid in cash
Experiment 4: Results
56%
83%
79%
96%
77%
72%
75%
89%
100%
 Ultimatum Game
B
A
B
 Reverse Impunity Game
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 0
B: 0
B
A
B
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 50
B: 0
 Although
• equal splits are almost never rejected
• equal splits are not expected to be rejected
 a non-neglible share of people behaves as if there is a
non-neglible chance that an equal split will be rejected.
 Ultimatum Game
B
A
B
 Reverse Impunity Game
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 0
B: 0
B
A
B
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 50
B: 0
 Behavior
• is not compatible with a consistency between strategies
and rational beliefs
• implies extreme risk aversion / pessimism when dealing
with people (strategic risk), as opposed to lotteries
(regular risk)
 Thank you.
 Ultimatum Game
B
A
B
 Reverse Impunity Game
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 0
B: 0
B
A
B
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 50
B: 0
 Behavior is not compatible with a consistency between
strategies and rational beliefs
Social preferences Errors
Risk Aversion
 Ultimatum Game
B
A
B
 Reverse Impunity Game
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 0
B: 0
B
A
B
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 50
B: 0
 Maximin *behavior*?
 Von Neumann & Morgenstern: “...the rules of rational behaviour
must provide definitely for the possibility of irrational conduct on the
part of others”
 Selten (2001): vNM “do not pin down what to be expected from
other players”, in vNM’s pre-Nash concept “players concentrate on
what can be assured in a game”
 Ultimatum Game
B
A
B
 Reverse Impunity Game
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 0
B: 0
B
A
B
A: 80
B: 20
A: 0
B: 0
A: 50
B: 50
A: 50
B: 0
 Strategic ambiguity aversion
• Gilboa, Schmeidler – non-additive beliefs / capacities
• Chateauneuf, Eichberger & Grant (2007) –
non-extreme-outcome additive capacities
Vi  si ,  i , i ,  i    i  i max ui  si , si   1   i  min ui  si , si    1   i  Ε i ui  si , si 
 s i S i

s i S i
• Eichberger, Kelsey (, Schipper) (2010, 2008, 2009) –
compare to empirical / experimental data