Shown in #MCDE2014 Magno Coloquio de Doctorantes en Economía 2014, in Mexico City, November 2014. Rights Reserved to Escuela Superior de Economía, Instituto Politécnico Nacional

1. Stochastic Differential Games:
An Application to Pension
Funds under Adverse Selection
Mario A. García-Meza
José Daniel López-Barrientos
Magno Coloquio de Doctorantes en Economía

2. Overview
Adverse Selection
Stochastic Differential
Games
An Application to
Pension Funds:
Separating Equilibrium
Conclusions

3. ADVERSE SELECTION

5. STOCHASTIC
DIFFERENTIAL
GAMES

6. dx(t) = b(x(t), u1(t), u2(t))dt + (x(t))dW(t)

7. 8i6= j, xi 2 Si : fi(x⇤i
, x⇤j
) $ fi(xi, x⇤j
)

9. Lu1,u2⌫(x) :=
Xn
i=1
bi(x, u1, u2)
@⌫
@xi
(x) +
1
2
Xn
i,j
aij(x)
@2⌫
@xi@xj
(x)
dx(t) = b(x(t), u1(t), u2(t))dt + (x(t))dW(t)

10. rk(x, ⇡1,⇡2) :=
Z
U2
Z
U1
rk(x, u1, u2)⇡1(du1|x)⇡2(du2|x)
J`T
(x, ⇡1,⇡2) := E⇡1,⇡2
x [
Z T
0
rk(x(t),⇡1,⇡2)dt]
Jk(x, ⇡1,⇡2) := lim
T!1
sup
1
T
Jk
T (x, ⇡1,⇡2)

11. J1(⇡⇤1,⇡⇤2) ! J1(⇡1,⇡⇤2) for every ⇡1 2 ⇧1
J2(⇡⇤1,⇡⇤2) ! J2(⇡⇤1,⇡2) for every ⇡2 2 ⇧2

12. Optimal Response
⇡1 = ”I know”
⇡1 = silence
⇡1 = ”Who doesn’t?”
⇡1 = ”I love you too”
⇡1 = ”Thank you”
⇡1 = ”I love me too”
⇡2 = ”I love you”

13. Optimal Response
J1(⇡⇤1,⇡2) = sup
⇡12⇧1
J1(⇡1,⇡2)
J2(⇡1,⇡⇤2) = sup
⇡22⇧2
J2(⇡1,⇡2)

14. Optimal equations of
Average Payoff
J1 = r1(x, ⇡⇤1,⇡2) + L⇡⇤1,⇡2
h1(x)
= sup
2¯U
1{r1(x, , ⇡2) + L,⇡2
h1(x)}
J2 = r2(x, ⇡⇤1,⇡⇤2) + L⇡⇤1,⇡2⇤h2(x)
= sup
2¯U
2{r2(x, ⇡⇤1, ) + L⇡⇤1, h2(x)}

16. dx(t) = (μx(t) + u1(t) u2(t))dt + dW(t)
x(0) = x
r1(x, u1, u2) = x
r2(x, u1, u2) = u2
Contribution Strategy
u1, u2 2 [0, d]
Withdrawal Strategy

17. J1= sup
2P(U1)

x + (μx + !(x) ⇡2(·|x))h01(x) +
1
2
#2h00 1 (x)
= sup
[!(x)h01(x)] + [x(1 + μ) ⇡2(·|x)]h01(x) +
2P(U1)
1
2
#2h00 1 (x)
J2= sup
2P(U2)

(x) + (μx + ⇡(·|x) (x))h02(x) +
1
2
#2h00 2 (x)
= sup
[ (x)(1 h02(x))] + (μx + ⇡1(·|x))h02(x) +
2P(U2)
1
2
#2h00 2 (x)

18. h01(x)
1 1 1
0
0
h02(x)
(0(x), d(x)) (0(x), u(x)) (0(x), 0(x))
(d(x), d(x)) (d(x), u(x)) (d(x), 0(x))

19. h01(x)
1 1 1
0
Contribute
Nothing, Withdraw
all possible
Contribute
nothing,withdraw
arbitrary amount
Contribute
Living off my
nothing, withdraw
rents
nothing
0
Contribute all
possible,withdraw
all possible
Contribute all
possible,withdraw
arbitrary amount
Contribute all
possible,withdraw
nothing
h02(x)
Abandon ship!
New Money Buy and Hold

21. Conclusions
An SDG with additive structure and average payoff can
yield Nash equilibria with deterministic strategies that
construct a separating equilibrium for pensions funds,
thus solving the adverse selection problem
This work can be extended still more by adding for
example (1) The control set from fund manager(s), (2)
The agent has to optimize through the selection of an
optimal portfolio

22. References
Pension Funds and Adverse Selection
Akerlof (1970). The Market for “lemons”: Quality uncertainty
and the market mechanisms. The quarterly journal of
economics, 488-500.
Blake, D. (1999). Annuity markets: Problems and Solutions.
Geneva Papers on Risk and Insurance. Issues and practice,
358-375.
Model for Stochastic Differential Game
Escobedo-Trujillo, B. A., Lopez-Barrientos, J.D. (2014)
Nonzero- Sum stochastic Differential Games with Additive
structure and average payoff. Journal of Optimization Theory
and Applications.