Reward and Punishment in a Regime Change
Game
Stephen Morris and Mehdi Shadmehr
Columbia Conference on Political Economy, December 2016
Question
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citizens choose among a repertoire of contentious
performances (Tilly 08):
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public meetings, boycotts, strikes, marches, demonstrations,
sit-ins, freedom rides, street blockades, suicide bombings,
assassination, hijacking, and guerrilla war
but what contentious performances can be elicited from
supporters and how?
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pleasure in agency: "the positive e¤ect associated with
self-determination, autonomy, self-esteem, e¢ cacy, and pride
that come from the successful assertion of intention" (Wood
03)
"depends on the likelihood of success, which in turn depends
on the number participating... yet the pleasure in agency is
undiminished by the fact that one’s own contribution to the
likelihood of victory is vanishingly small"
Question
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transformational leadership: ability to create inspirational
motivations through a variety of psychological mechanisms
(Burns 78)
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our question:
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sociology: leaders raise participation by "identi…cation,
idealization, elevation of one or more values..." (Snow et al 89)
politics: "people-oriented leaders are those who inspire people,
give them a sense of identity..." (Goldstone 01)
what happens when transformative leaders manipulate pleasure
in agency in order to in‡uence activists’choice among a
repertoire of contentious performances?
our answer (in a stylized model)
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"the organization of the revolutionaries must consist …rst and
foremost of people who make revolutionary activity their
profession"....while others (workers) engage in various levels of
contentious activities (Lenin 1902)
Key Tradeo¤s
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if activists ("citizens") are heterogeneous, then there would be
trade-o¤s in manipulating pleasure in agency ("bene…ts");
higher bene…ts from intermediate levels of contentious
performances ("e¤ort") will...
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encourage those who would otherwise have chosen low e¤ort
to choose intermediate e¤ort
discourage those who might otherwise have chosen high e¤ort
even if citizens are not heterogenous, at the margin where the
revolution is occuring, the citizenry will have heterogeneous
beliefs about the likelihood of success
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if everyone thought revolution was going to succeed, everyone
would choose maximum e¤ort and the revolution would
succeed
if everyone thought revolution was going to fail, everyone
would choose minimum e¤ort and the revolution would fail
Optimal Reward Schemes
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we will consider a situation where there is an upper bound on
bene…ts...
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the upper bound on bene…ts will imply an upper bound on
e¤ort (what would be chosen if you assigned probability 1 to
success)
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we will identify an optimal reward scheme (mapping from
e¤ort to bene…ts)..
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there will be a critical level of e¤ort at which citizens will get
the maximal bene…t...
bene…ts will smoothly decline for lower e¤orts
Model of Coordination...
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continuous action regime change game:
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citizens make a continuous e¤ort decision
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revolution succeeds if aggregate e¤ort exceeds a threshold
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small uncertainty about threshold ) unique equilibrium
("global game")
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at critical threshold, there will be uniform distribution among
citizens of the probability of success
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citizens trade o¤ costs and success contingent reward scheme
....and Screening
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leader can pick optimal reward scheme
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screening problem because of heterogeneous beliefs about the
probability of success
Methodological Contribution
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continuous action regime change game (Guimaraes-Morris 07
is unique applied precursor?)
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global game with screening
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non-standard screening problem: role of maximum choice
variable
Extensions and Open Questions
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designing costs instead of bene…ts (same principle but type
dependent participation constraints)
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a game between players choosing costs and bene…ts
respectively
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more standard screeing problems
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philanthropy and other applications
Model with Exogenous Bene…ts and Complete Information
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continuum of citizens with each choosing ei 2 [0, 1]
revolution succeeds if
strength"
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θ, where θ is "regime
C (0) = C 0 (0) = 0, C 0 (e ), C 00 (e ) > 0 for e > 0
contingent bene…t of e¤ort B (e )
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ei di
uncontingent cost of e¤ort C (e )
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Z
B (0) = 0, B 0 (e ) > 0
joint restrictions:
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C (e )
B (e )
strictly increasing, B 00 (e ) < C 00 (e ) for all e and
B (e ) < C (e ) for some e
Equilibrium with Exogenous Bene…ts and Complete
Information
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e¤ort e if you are sure revolution will occur solves
B 0 (e ) = C 0 (e )
RESULT 1: There are smallest and largest Nash equilibria where
citizens exert e¤ort 0 and e respectively
Model with Exogenous Bene…ts and Incomplete
Information
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state θ g ( ), citizen i observes xi = θ + σεi , where each εi
is independent distribution according to f ( )
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we will now care about
e (p ) = arg max pB (e )
e
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and we will study what happens as σ ! 0
C (e )
Equilibrium with Exogenous Bene…ts and Incomplete
Information
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this is a global game with continuum actions
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there is a unique equilibrium in the incomplete information
game: Frankel, Morris and Pauzner (2003)
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if (but only if) we have a "regime change game", there is a
very nice characterization of that unique equilibrium
Equilibrium with Exogenous Bene…ts and Incomplete
Information
RESULT 2: The unique equilibrium will be a "threshold
equilibrium" with a critical level of regime strength θ such that in
limit of the incomplete information game, revolution succeeds if
and only if θ > θ where
θ =
Z1
e (p ) dp
1
C 0 (e )
B 0 (e )
(*)
p =0
and so
θ =
Ze
e =0
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de
Follows from argument of Guimaeres and Morris (2006)
Statistical Fact
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…x any b
θ
at b
θ, a citizen with signal x will assign some probability p to
θ b
θ
STATISTICAL FACT: the distribution of p’s in the population
is uniform
Statistical Fact Proof
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…x any b
θ and p
player assigns probability p to θ
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b
θ
the probability an observer assigns to x
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b
x
ε b
θ or ε x
θ
this occurs with probability F x
b
θ when x
x
x if b
θ+ε
x ; prob F x
b
θ=F
x is F x
b
θ event
so probability observer assigns to p is F F
1
(p ) = p
1
(p )
b
θ
Endogenous Bene…ts
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Leader chooses B : R+ ! [0, M ] to maximize the probability
of revolution, i.e., θ
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motivation: see introduction
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modelling choices: many alternatives
Recall Two Insights (from introduction)
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if supporters ("citizens") are heterogeneous, then there would
be trade-o¤s in manipulating bene…ts; higher bene…ts from
intermediate levels of e¤ort will...
I
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I
encourage those who would otherwise have chosen low e¤ort
to choose intermediate e¤ort
discourage those who might otherwise have chosen high e¤ort
even if citizens are not heterogeneous, at the margin where
the revolution is occuring, the citizenry will have
heterogeneous beliefs about the likelihood of success
I
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if everyone thought revolution was going to succeed, everyone
would choose maximum e¤ort and the revolution would
succeed
if everyone thought revolution was going to fail, everyone
would choose minimum e¤ort and the revolution would fail
Insights Applied
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Leader chooses B to maximize
θ =
Z1
e (p ) dp
p =0
subject to e being the e¤ort levels induced by B
(*)
Screening Problem
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Standard revelation principle / optimal screening argument
can re-write problem as:
MAXIMIZATION PROBLEM: Leader chooses B, e to maximize
θ =
Z1
e (p ) dp
(*)
p =0
subject to...
pB (p )
C (e (p ))
0
pB (p )
C (e (p ))
pB p 0
B (p ) 2 [0, M ]
C e p0
Buyer Seller Screening Problem
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with C ( ) is linear, as if a buyer of type p is buying a good of
quality B from a seller for price e, where only constraint on B
is its maximum
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could solve this problem with essentially standard pointwise
methods
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non-linear C ( ) creates complications
Main Result
RESULT 3: Suppose that C is strictly convex. Then the optimal
B ( ) is continuous, weakly increasing and - for some 0 < b
e<1strictly convex on [0, b
e ] and equal to M above b
e . The optimal e is
continuous, strictly increasing on [0, 1/2] and equal to b
e constant
on [1/2, 1].
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Why bunching?
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Suppose that 0 < B 0 (e ) < C 0 (e ) on an interval [e1 , e2 ]; no
one will choose e¤ort in the interval [e1 , e2 ]
Must have B 0 (e ) C 0 (e ) for unless B 0 is zero or e = e
Example: Optimal Bene…ts
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M = 1 and c (e ) = e n for some n
1
Example: Optimal E¤ort
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M = 1 and c (e ) = e n for some n
1
Main Result
RESULT 3: Suppose that C is strictly convex. Then the optimal
B ( ) is continuous, weakly increasing and - for some 0 < b
e<1strictly convex on [0, b
e ] and equal to M above b
e . The optimal e is
continuous, strictly increasing on [0, 1/2] and equal to b
e constant
on [1/2, 1].
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Why bunching?
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Suppose that 0 < B 0 (e ) < C 0 (e ) on an interval [e1 , e2 ]; no
one will choose e¤ort in the interval [e1 , e2 ]
Must have B 0 (e ) C 0 (e ) for unless B 0 is zero or e except at
the top
Bunching at the top
Extensions and Open Questions
I
designing costs instead of bene…ts (same principle but type
dependent participation constraints)
I
a game between players choosing costs and bene…ts
respectively
I
more standard screeing problems
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philanthropy and other applications
Conclusion
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Methodology:
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incorporating continuous actions and screening into models of
revolution (many novel comparative statics we could study)
Substantive:
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subtle tradeo¤s in design of optimal pleasure in agency
in support of the vanguard