Stochastic Differential Games: An Application to Pension Funds under Adverse SelectionMario A. García-Meza José Daniel López-Barrientos

Magno Coloquio de Doctorantes en Economía

OverviewAdverse Selection

Stochastic Differential Games

An Application to Pension Funds: Separating Equilibrium

Conclusions

ADVERSE SELECTION

STOCHASTIC DIFFERENTIAL GAMES

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

8i 6= j, xi 2 Si : fi(x⇤i , x

⇤j ) � fi(xi, x

⇤j )

Lu1,u2⌫(x) :=

nX

i=1

bi(x, u1, u2)@⌫

@xi(x) +

1

2

nX

i,j

a

ij(x)@

2⌫

@xi@xj(x)

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

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

[

ZT

0r

k

(x(t),⇡1,⇡

2)dt]

J

k(x,⇡1,⇡

2) := limT!1

sup1

T

J

kT (x,⇡

1,⇡

2)

J1(⇡⇤1,⇡⇤2

) � J1(⇡1,⇡⇤2

) for every ⇡1 2 ⇧

1

J2(⇡⇤1,⇡⇤2

) � J2(⇡⇤1,⇡2

) for every ⇡2 2 ⇧

2

Optimal Response

⇡2= ”I love you”

⇡1= ”I know”

⇡1 = silence

⇡1= ”Thank you”

⇡1= ”I love me too”

⇡1= ”I love you too”

⇡1= ”Who doesn’t?”

Optimal Response

J1(⇡⇤1,⇡2) = sup⇡12⇧1

J1(⇡1,⇡2)

J2(⇡1,⇡⇤2) = sup⇡22⇧2

J2(⇡1,⇡2)

Optimal equations of Average Payoff

J

1 = r1(x,⇡⇤1,⇡

2) + L⇡⇤1,⇡2

h1(x)

= sup�2U1

{r1(x,�,⇡2) + L�,⇡2

h1(x)}

J

2 = r2(x,⇡⇤1,⇡

⇤2) + L⇡⇤1,⇡2⇤

h2(x)

= sup 2U2

{r2(x,⇡⇤1, ) + L⇡

⇤1, h2(x)}

dx(t) = (µx(t) + u1(t)� u2(t))dt+ �dW (t)

x(0) = x

dx(t) = (µx(t) + u1(t)� u2(t))dt+ �dW (t)

x(0) = x

r1(x, u1, u2) = x

u1, u2 2 [0, d]

r1(x, u1, u2) = x

r2(x, u1, u2) = u2

Contribution Strategy

Withdrawal Strategy

J

1 = sup�2P(U1)

x+ (µx+ �(x)� ⇡

2(·|x))h01(x) +

1

2�

2h

001(x)

�

= sup�2P(U1)

[�(x)h01(x)] + [x(1 + µ)� ⇡

2(·|x)]h01(x) +

1

2�

2h

001(x)

J

1 = sup�2P(U1)

x+ (µx+ �(x)� ⇡

2(·|x))h01(x) +

1

2�

2h

001(x)

�

= sup�2P(U1)

[�(x)h01(x)] + [x(1 + µ)� ⇡

2(·|x)]h01(x) +

1

2�

2h

001(x)

J

2 = sup 2P(U2)

(x) + (µx+ ⇡(·|x)� (x))h0

2(x) +1

2�

2h

002(x)

�

= sup 2P(U2)

[ (x)(1� h

02(x))] + (µx+ ⇡

1(·|x))h02(x) +

1

2�

2h

002(x)

J

2 = sup 2P(U2)

(x) + (µx+ ⇡(·|x)� (x))h0

2(x) +1

2�

2h

002(x)

�

= sup 2P(U2)

[ (x)(1� h

02(x))] + (µx+ ⇡

1(·|x))h02(x) +

1

2�

2h

002(x)

h

01(x)

<1 1 >1

<0

>0

h

02(x)

(�0(x), �d(x)) (�0(x), �u(x)) (�0(x), �0(x))

(�d(x), �d(x)) (�d(x), �0(x))(�d(x), �u(x))

h

01(x)

<1 1 >1

<0Contribute

Nothing, Withdraw all possible

Contribute nothing,withdraw arbitrary amount

Contribute nothing, withdraw

nothing

>0Contribute all

possible,withdraw all possible

Contribute all possible,withdraw arbitrary amount

Contribute all possible,withdraw

nothing

h

02(x)

Abandon ship!

Buy and HoldNew Money

Living off my rents

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

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.

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.

ReferencesPension Funds and Adverse Selection

Model for Stochastic Differential Game