Abstract
In this paper we define and solve a ‘robust game design’ problem that could be used to assess the fair sharing of the abatement burden among the 28 EU countries in their coming climate negotiations. The problem consists in finding a distribution of a global ‘safety emission budget’ for the panning period 2010–2050, among the 28 countries in such a way to obtain a balanced relative loss of welfare (computed in percent of the discounted consumption in the reference case) when the countries supply strategically their permit endowment on a permit trading system with full banking and borrowing. We assume that the countries play a noncooperative game, where the payoffs are constituted of the gains from the terms of trade plus the gains in the permit trading and minus the abatement cost, expressed as the compensative variation of income. These payoff functions are estimated from an ensemble of numerical simulations of a detailed CGE model, GEMINI-E3 representing the economic interactions among the 28 EU countries. To deal with the uncertainty introduced by the statistical emulation technique we propose to use the concept of robust equilibrium, where the results of robust optimization are exploited in the definition of an equilibrium solution, when the payoff is subject to uncertainties. A numerical illustration is performed and an interpretation of the impact of the robustification approach on the solution of the game design problem is provided.
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- 1.
Note that, as the second derivative of the AC appears in the mathematical game formulation, we have thus imposed a convexity constraint on this second derivative in the regression model in order to ensure the convexity of the overall problem. Moreover, polynomial forms of lower degree have been tested but resulting in worse estimation quality.
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Acknowledgements
The research leading to these results has received funding from Qatar National Research Fund under Grant Agreement no 6-1035-512.
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Appendix
Appendix
15.1.1 GAMS Code of the Robust Game
We report below the GAMS implementation code of the robust formulation to the game. This model is solved with the PATH solver (Ferris and Munson 2000).
sets
J PLAYERS
T PERIODS 2020 TO 2050
;
Table
EX(J,T) EXCHANGE RATES
GTT(J,T) GAINS FROM TERMS OF TRADE LINEAR TERM
EB(J,T) BAU emissions
A0(J,T) COEFF. CONST. MAC FUNCTION
A1(J,T) COEFF. LIN. MAC FUNCTION
A2(J,T) COEFF. QUAD. MAC FUNCTION
A3(J,T) COEFF. CUB. MAC FUNCTION
A1_var(J,T) VARIABILITY OF COEFF. LIN. MAC FUNCTION
A2_var(J,T) VARIABILITY OF COEFF. QUAD. MAC FUNCTION
A3_var(J,T) VARIABILITY OF COEFF. CUB. MAC FUNCTION
;
scalar
BUD GLOBAL EMISSION BUDGET
beta DISCOUNT FACTOR
;
parameter
BS(J) SHARES OF EM. BUDGET
;
positive variable
a(T,J) ALLOWANCES
e(T,J) EMISSION LEVELS
q(T,J) ABATEMENT LEVELS
P(T) PERMIT PRICES
nu(J) MULTIPLIER ALLOWANCE
;
variable
DAcost(T,J) MARGINAL ABATEMENT COSTS
DDAcost(T,J) SECOND DERIVATIVE ABATEMENT COST
totA(T) TOTAL ALLOWANCES
totE(T) TOTAL EMISSIONS
cE CUMULATIVE EMISSIONS
dP(T) DIFF PRICE
dE(T,J) DIFF EMISS
TR(T,J) NET TRANSFERS
;
equations
* CHECK THAT THE SOLUTION USES THE BUDGET SHARE
BcumA(j).. BUD*BS(J)- (10*sum(T, a(T,J))) =g= 0;
* DEFINES TOTAL ALLOWANCES AT T
EtotA(T).. totA(T) - sum(J,a(T,J)) =e= 0;
* DEFINES TOTAL EMISSIONS
EtotE(T).. totE(T) - sum(J,e(T,J)) =e= 0;
* DEFINES EMISSIONS FROM ABATEMENT AT T
EQe(T,J).. -EB(J,T)+ e(T,J) + q(T,J) =e= 0;
* DEFINES TOTAL EMISSIONS
EQce.. cE =e= 10*sum(T, totE(T));
* DEFINES MAC
EQDAcost(T,J).. DAcost(T,J) - (A1(J,T)*q(T,J)+
A2(J,T)*q(T,J)**2+A3(J,T)*q(T,J)**3)/EX(J,T)
- k2*sqrt(abs(A1_var(J,T)*q(T,J))**2 +
abs(A2_var(J,T)*q(T,J)**2)**2 +abs(
A3_var(J,T)*q(T,J)**3)**2)/EX(J,T) =e= 0;
* DEFINES MINUS DERIVATIVE OF MAC
EDDAcost(T,J).. DDAcost(T,J)+(A1(J,T)+2*A2(J,T)
*q(T,J)+3*A3(J,T)*q(T,J)**2)/EX(J,T) + k2*(2*
abs(A1_var(J,T))**2*q(T,J)+4*abs(A2_var(J,T))**2
*q(T,J)**3 + 6*abs(A3_var(J,T))**2*q(T,J)**5)
/ (2*EX(J,T)*sqrt( abs(A1_var(J,T)*q(T,J))**2
+abs(A2_var(J,T)*q(T,J)**2)**2
+abs(A3_var(J,T)*q(T,J)**3)**2)) =e= 0;
* DEFINES DERIVATIVE OF MARKET PRICE WRT ALLOWANCE
EdP(T).. dP(T) - 1/sum(J,1/DDAcost(T,J)) =e= 0;
* DEFINES DERIVATIVE OF EMISSION WRT ALLOWANCE
EdE(T,J).. dE(T,J) - 1/sum(I,DDAcost(T,J)
/DDAcost(T,I)) =e= 0;
* DEFINES PSEUDO-GRADIENT OF PAYOFFS W.R.T.
* ALLOWANCES, TAKING INTO ACCOUNT EFFECTS ON PRICES.
PSGRAD(T,J).. -(1+beta)**(-10*(ord(T)-1))
*(DAcost(T,J)
+ dP(T)*(a(T,J)-e(T,J)) - GTT(J,T))
+ Nu(J)
* (sqrt(var*sum(I,sum(TI,(10*a(TI,I))**2)))
+ a(T,J)*k*var)
=e= 0;
* MARKET CLEARS (TOTAL EMISSIONS EQUAL TOTAL
* ALLOWANCES AT T)
MARKETC(T).. totA(T)- totE(T) =e= 0;
* PRICE IS EQUAL TO MAC
MAXPRO(T,J).. DAcost(T,J) - P(T) =e= 0;
* TRANSFERS
TRANS(T,J).. TR(T,J)-P(T)*(a(T,J)-e(T,J)) =e= 0;
model robust-game
/
BcumA.Nu,
EtotA.totA,
EtotE.totE,
EQe.q,
EQce.cE,
EQDAcost.DAcost,
EDDAcost,
EdP.dP,
EdE.dE,
PSGRAD.a,
MARKETC.P,
MAXPRO.e,
TRANS.TR
/;
option mcp=path;
solve robust-game using mcp;
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Babonneau, F., Haurie, A., Vielle, M. (2016). A Robust Noncooperative Meta-game for Climate Negotiation in Europe. In: Thuijsman, F., Wagener, F. (eds) Advances in Dynamic and Evolutionary Games. Annals of the International Society of Dynamic Games, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28014-1_15
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