17th ISA World Congress July 17 2010,
RC45
ISA2010 RC45.10
17 July, 2010
Purpose of the Presentation
„
An Initial Condition Game
of the Richardson Model
of Arms Races
To point out another type of micro-macro
linkage for social change
Complementing
Crucial
„
Atsushi ISHIDA
(Kwansei Gakuin University)
the Coleman’s boat
for historical sociology
To propose an “initial condition game” of the
Richardson’s arms races model, as a simple
example of the above framework
Interdependent
rational choice situation for
influential players on the premise of people’s
tendencies
1
Coleman’s Boat
2
Overturned Coleman’s boat
Macro Situatoin X(t)
Recognition of Situation
by Influential Players
Macro Situation X(t +1)
Rational Actions
by Actors
Recognition of
Situation by Actors
Macro Situation X(t)
Rational Actions by
Influential Players
Macro Situation X(t +1)
3
Unit Process
Unintended Outcome of Process
Micro process driven by
influential players
X(t -1)
4
X(t)
X(t + 1)
Macro changes
Micro process driven by
ordinary people
for influential players
„ Essential uncertainty of process
„ Limits of recognition and rationality
„ Interdependency of rational choices by
players
„
5
Game theory + Differential Equation Model
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17th ISA World Congress July 17 2010,
RC45
Richardson’s Arms Races Model
An Example of Stream of x, y
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Armament of the country decided through a
democratic process
x, y : quantity of arms in the country X and Y
k, l : defense (rival) coefficient k > 0, l > 0
α, β : fatigue coefficients α > 0, β > 0
g, h : grievances factors
„
15
10
x' = k y − α x + g
y' = l x − β y + h
5
0
7
0
5
10
15
Initial Condition Game
Initial Condition Game
Symmetric condition, same values of
parameters
„ Assume k > α, g < 0 (k = 2, a = 1, g = − 5)
„
„
: the limit value (maximum value) of x, y
Each player chooses an initial value x0, y0 from [0,
xm], [0, ym], respectively.
armament race, tending toward either
disarmament or war
„
x' = k y − α x + g
y' = k x − α y + g
Utility function
U X ( x, y ) = s X ( x + 1)γ X ( y + 1)1−γ X
U Y ( x, y ) = sY ( x + 1)γ Y ( y + 1)1−γ Y
Two players
Leaders
Strategies
xm, ym
Unstable
„
8
20
or Social planners representing X and Y
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Initial Condition Game
„
Unit Process of Arms Races
Initial Condition Game
(I.C.G)
Nash Equilibriums
(
( ) )≥ U
X
( lim x(t, x ) )
( ) )≥ U
Y
( lim x(t, x ) )
∀x0 ∈ [0, xm ], U X lim x t , x*0
(
t →∞
∀y0 ∈ [0, ym ], U Y lim x t , x*0
t →∞
10
t →∞
t →∞
0
(x(t -1),y(t -1))
0
Macro changes
(x(t),y(t))
(x(t+1),y(t+1))
x' = k y − α x + g
y' = l x − β y + h
where x = ( x, y )
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Differential Equation Model
(D.E.M)
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17th ISA World Congress July 17 2010,
RC45
xm = 15, ym = 15
Exp. 1: Coincidence of Interest
„
The representative of each country wants to
make disarmament in his/her own country
sX = − 1, sY = − 1, γX = 1, γy = 0
Best Response
for X
U X = −( x + 1)
U Y = −( y + 1)
Nash Equilibriums
Best Response
for Y
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Exp. 2: Conflict of Interest in
Interdependent situation
xm = 15, ym = 15
„
The representative of country X wants to make
disarmament in the both country simultaneously
„
The representative of country Y wants to
enhance further arms race involving the both
country simultaneously
Best Response
for X
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No Nash Equilibriums
Best Response
for Y
sX = − 1, sY = 1, γX = 1/2, γy = 1/2
U X = −( x + 1)1/ 2 ( y + 1)1/ 2
U Y = ( x + 1)1/ 2 ( y + 1)1/ 2
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xm = 10, ym = 10
16
xm = 7, ym = 15
Nash Equilibrium
(0, ym)
Nash Equilibrium
(0, 2g /(α− k))
Best Response
for X
Best Response
for Y
Best Response
for X
Best Response
for Y
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17th ISA World Congress July 17 2010,
RC45
Exp. 2: Conflict of Interest in
Interdependent situation
„
Conclusion
Existence of Equilibriums depends on (xm, ym)
„
2g ⎞
⎛
( xm < y m ) ∧ ⎜ y m >
⎟ ⇒ (0, ym )
α −k ⎠
⎝
ym =
„
Two types of micro-macro process as a unit
process
Important
especially for historical analysis
consequences
Unintended
„
2g
2g ⎞
⎛
⇒ ⎜ 0,
⎟
α −k ⎝ α −k⎠
Initial condition game
Game
theoretic structure over the differential equation
flow of event
If Y has sufficiently large amount of potential
productivity, the representative of Y can achieve
the goal to some extent, by putting maximum
amount of armament as the initial condition
„
Future tasks
Related
model (Brito 1972; Intriligator 1975; Gillespie et
al. 1977)
Changes in parameters in D.E.M. by I.C.G.
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References
„
„
„
„
„
Richardson, L. F. 1939. Generalized Foreign Politics: A
Study in Group Psychology. British Journal of Psychology.
Monograph Supplement No. 23, Cambridge : Cambridge
University Press.
Richardson, L. F. 1960. Arms and Insecurity: A
Mathematical Study of the Causes and Origins of War.
Edited by Nicolas Rashevsky and Ernesto Trucco.
Pittsburgh: Boxwood Press.
Brito, D. L. 1972. "A Dynamic Model of an Armaments
Race." International Economic Review 13:359-375.
Intriligator, M. D. 1975. "Strategic Considerations in the
Richardson Model of Arms Races." Journal of Political
Economy 83:339-354.
Gillespie, J. V., D. A. Zinnes, G. S. Tahim, P. A. Schrodt,
and R. M. Rubison. 1977. "An Optimal Control Model of
Arms Races." American Political Science Review 71:226.
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