ps10_s_mn.gms : Parts Supply Problem w/ 10 Types w/ Random p(i)

**Description**

Hideo Hashimoto, Kojun Hamada, and Nobuhiro Hosoe, "A Numerical Approach to the Contract Theory: the Case of Adverse Selection", GRIPS Discussion Paper 11-27, National Graduate Institute for Policy Studies, Tokyo, Japan, March 2012. http://www.grips.ac.jp/r-center/en/discussion_papers/11-27/ Keywords: nonlinear programming, contract theory, principal-agent problem, adverse selection, parts supply problem

**References**

- Hashimoto, H, Hamada, K, and Hosoe, N, A Numerical Approachto the Contract Theory: The Case of Adverse Selection. GRIPS Discussion Papers, National Graduate Institute for Policy Studies, 2012.
- Itoh, H, A Course in Contract Theory. Yuhikaku, Tokyo, 2003.

**Small Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** ps10_s_mn.gms

```
$title Parts Supply Problem w/ 10 Types w/ Random p(i) (PS10_S_MN,SEQ=369)
$onText
Hideo Hashimoto, Kojun Hamada, and Nobuhiro Hosoe, "A Numerical Approach
to the Contract Theory: the Case of Adverse Selection", GRIPS Discussion
Paper 11-27, National Graduate Institute for Policy Studies, Tokyo, Japan,
March 2012.
http://www.grips.ac.jp/r-center/en/discussion_papers/11-27/
Keywords: nonlinear programming, contract theory, principal-agent problem,
adverse selection, parts supply problem
$offText
option limCol = 0, limRow = 0, solPrint = off;
Set
i 'type of supplier' / 0*9 /
t 'no. of Monte-Carlo draws' / 1*1000 /;
Alias (i,j);
Parameter
theta(i) 'efficiency'
pt(i,t) 'probability of type'
p(i) 'probability of type';
theta(i) = ord(i)/card(i);
* Generating probability
loop(t, pt(i,t) = uniform(0,1););
pt(i,t) = pt(i,t)/sum(j,pt(j,t));
* pt(i,"1") = 1/card(i);
Parameter
F(i,t) 'cumulative probability (Itho p. 42)'
noMHRC0(i,t) 'no MHRC combination between i and i-1'
* (MHRC: monotone hazard rate condition)
noMHRC(t) '>=1: no MHRC case';
F(i,t) = sum(j$(ord(j) <= ord(i)), pt(j,t));
noMHRC0(i,t) $(ord(i) < card(i)) = 1$(F(i,t)/pt(i+1,t) < F(i-1,t)/pt(i,t));
noMHRC(t)$(sum(i, noMHRC0(i,t)) >= 1) = 1;
Scalar ru 'reservation utility' / 0 /;
* Definition of Primal/Dual Variables
Positive Variable
x(i) "quality"
b(i) "maker's revenue"
w(i) "price";
Variable Util "maker's utility";
Equation
obj "maker's utility function"
rev(i) "maker's revenue function"
pc(i) "participation constraint"
licd(i) "incentive compatibility constraint"
licu(i) "incentive compatibility constraint"
ic(i,j) "global incentive compatibility constraint"
mn(i) "monotonicity constraint";
obj.. Util =e= sum(i, p(i)*(b(i) - w(i)));
rev(i).. b(i) =e= x(i)**(0.5);
pc(i).. w(i) - theta(i)*x(i) =g= ru;
licd(i).. w(i) - theta(i)*x(i) =g= w(i+1) - theta(i)*x(i+1);
licu(i).. w(i) - theta(i)*x(i) =g= w(i-1) - theta(i)*x(i-1);
ic(i,j).. w(i) - theta(i)*x(i) =g= w(j) - theta(i)*x(j);
mn(i).. x(i) =g= x(i+1);
* Setting Lower Bounds on Variables to Avoid Division by Zero
x.lo(i) = 0.0001;
Model
SB_lic / obj, rev, pc, licd /
SB_lic2 / obj, rev, pc, licd, mn /;
* Options to solve models quickly
SB_lic.solveLink = 5;
SB_lic2.solveLink = 5;
Parameter
Util_lic(t) 'util solved w/o MN'
Util_lic2(t) 'util solved w/ MN'
Util_gap(t) 'gap between these two util'
x_lic(i,t) 'x solved in w/o MN'
x_lic2(i,t) 'x solved in w/ MN'
MN_lic(t) 'monotonicity of x solved w/o MN'
MN_lic2(t) 'monotonicity of x solved w/ MN';
loop(t,
p(i) = pt(i,t);
* Solving the model w/o MN
solve SB_lic maximizing Util using nlp;
Util_lic(t) = util.l;
x_lic(i,t) = x.l(i);
MN_lic(t) = sum(i, 1$(round(x.l(i),10) < round(x.l(i+1),10)));
* Solving the model w/ MN
solve SB_lic2 maximizing Util using nlp;
Util_lic2(t) = util.l;
x_lic2(i,t) = x.l(i);
MN_lic2(t) = sum(i, 1$(round(x.l(i),10) < round(x.l(i+1),10)));
);
Util_gap(t) = 1$(round(Util_lic(t),10) <> round(Util_Lic2(t),10));
* Computing probability that MHRC and MN holds.
Parameter
p_noMHRC 'no MHRC case [%]'
p_noMN_lic 'no MN case [%]'
p_Util_gap 'no util-equality case [%]';
p_noMHRC = sum(t$(noMHRC(t) > 0), 1)/card(t)*100;
p_noMN_lic = sum(t$(MN_lic(t) > 0), 1)/card(t)*100;
p_Util_gap = sum(t$(Util_gap(t) > 0), 1)/card(t)*100;
display p_noMHRC, p_noMN_LIC, p_Util_gap;
* Generating CSV file for summary
File sol / solution_lic.csv /;
put sol;
sol.pc = 5;
sol.pw = 32767;
put "";
loop(i, put "pt(i,t)";);
put "" "" "" "";
loop(i, put "x: w/o MN";);
loop(i, put "x: w/ MN";);
put /;
put "";
loop(i, put i.tl;);
put ">=1: no MHRC" "Util: w/o MN" "Util: w/ MN" "Util_gap: =1: not equal";
loop(i, put i.tl;);
loop(i, put i.tl;);
put "MN_lic: >=1: no MN" "MN_lic2: >=1: no MN"/;
loop(t, put t.tl;
loop(i, put pt(i,t):10:5;);
put noMHRC(t) Util_lic(t):20:10 Util_Lic2(t):20:10 Util_gap(t);
loop(i, put X_lic(i,t););
loop(i, put X_lic2(i,t););
put MN_lic(t) MN_lic2(t)/;
);
put /;
```