Virtual implementation : simulation by prolog

23 Nov 2003

Kenryo INDO

Kanto Gakuen University

Game theory and mechanism design• Game theory has been widely used in economic /management scien

ces since von Neumann and Morgenstern’s work in 1947.• Mechanism design (or implementation) theory [4,5] is a subfield of g

ame theory which has been applied to the social choice theory and other mathematical economic theories including some management technologies such as auction, accounting and congestion control.

• Game theorists (or their role as the social planner) tried to achieve to match the given desirable goal for the model of society (i.e., the social choice rule (SCF) for small sized team or market economy) with their autonomous, rational, selfish, or opportunistic behaviors, i.e., the set of Nash equilibrium, of the individual members.

• They found implementation theorems and developed game forms in order for the above objective. Lottery (or stage) mechanism with small fines [1—4] is known as the most powerful in that it can implement almost any SCF. Whereas other non-lottery (or single-stage) mechanisms are required so that the SCF satisfies monotonicity and if you further want to qualify unique equilibrium then the impossibility results in general unrestricted environment (i.e., only dictatorial or constant SCFs are implementable).

Logic programming and Prolog• We will utilize Prolog, a logic programming technique, in order for mo

deling game theoretic ( autonomous, and rational) agents and mechanisms for the coordination of agents society.

• Prolog is an abbreviation of “programming language in logic.”• The standard syntax is called Edinburgh-prolog or ISO-prolog.• A sentence of Prolog code is a definite clause which is the pair of hea

d, goal :- body, which can be seen as a disjunctive with one positive literal (i.e., Horn clause), `goal Or negation of body’ in logic.

• Goal of a clause is a predicate with some arity, and body is predicates separated with comma which is the sufficient condition of the goal. – Ex. city(X):- ancient_capital(X,Japan), X \= nara.

• Prolog is driven by the SLD-resolution technique that is developed in the joint-field of natural language processing and automated theorem proving systems.

an agency model (mere descriptive)

% a descriptive model of agency%--------------------------------

member_of_agency(manager).member_of_agency(worker1).member_of_agency(worker2).

possible_result_of_business(success(1)).possible_result_of_business(success(2)).possible_result_of_business(fail).

possible_decision_of_manager(reward(a)).possible_decision_of_manager(reward(b)).

possible_level_of_effort_of_worker(high).possible_level_of_effort_of_worker(low).

a game theoretical modeling by prolog

% the base model of agency%------------------------------------------------------game(agency(0), players([worker1,worker2]), acts([report(S1),report(S2)]), payoffs(Y) ):- member([S1,S2,Y], [ [success1,success1,[180,120]], [success1,success2,[0,0]], [success2,success1,[0,0]], [success2,success2,[120,180]] ] ).

Nash equilibrium and best response

best_response(G,J,N,S,P):- game(G, players(N),acts(S),payoffs(P) ), member(J,N), \+ defeated_by(G,J,N,[S,P],_).

nash(G,J,N,S,P):- best_response(G,J,N,S,P).

nash(G,players(N),acts(S),payoffs(P)):- game(G, players(N),acts(S),payoffs(P) ), \+ ( member(J,N), \+ best_response(G,J,N,S,P) ).

mutate(G,S,J,S1):- game(G,players(N),acts(S),_), game(G,players(N),acts(S1),_), subtract(N,[J],NJ),%write(remains(NJ)),nl, forall(member(J1,NJ), ( nth1(K,N,J1),%write(j1(J1,k(J))), nth1(K,S,SK),%write(s(SK)), nth1(K,S1,SK1),%write(s1(SK1)),nl, SK = SK1 )).defeated_by(G,J,N,[S,P],[S1,P1]):- game(G,players(N),acts(S),payoffs(P)), nth1(K,N,J), ( mutate(G,S,J,S1), game(G,players(N),acts(S1),payoffs(P1)), nth1(K,P1,PK1), nth1(K,P,PK), PK < PK1 ).

The Nash equilibria of four: two pair of truth-telling and of joint-fraud

?- nash(A,B,C,D).

A = agency(0)B = players([worker1, worker2])C = acts([report(success1), report(success1)])D = payoffs([180, 120]) ;

A = agency(0)B = players([worker1, worker2])C = acts([report(success2), report(success2)])D = payoffs([120, 180]) ;

A = agency(1, true_state(success1))B = players([worker1, worker2])C = acts([report(success1), report(success1)])D = payoffs([361, 120]) ;

A = agency(1, true_state(success2))B = players([worker1, worker2])C = acts([report(success2), report(success2)])D = payoffs([301, 180]) ;

No

a simple mechanism with large aux. reward

game(parameter,true_state,nature,S):- member(S,[success1,success2]).

game(parameter,auxiliary_reward1,worker1,181).

game(agency(1,true_state(S)), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([Y1,Y2]) ):- game(agency(0), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([X1,X2]) ), game(parameter,true_state,nature,S), game(parameter,auxiliary_reward1,worker1,M), (R1 =S -> Bonus =M ; Bonus =0), Y1 is X1 + Bonus, Y2 = X2.

% a run.

?- nash(agency(1,S),B,C,D).

S = true_state(success1)

B = [worker1, worker2]

C = [report(success1), report(success1)]

D = [361, 120] ;

S = true_state(success2)

B = [worker1, worker2]

C = [report(success2), report(success2)]

D = [301, 180] ;

No

?-

Virtual implementation in iteratively undominated strategies under com

plete information• A-M mechanism was developed by Abreu and Matsushima [1,2] for two or

more agents is the stage game with small fines which yields a lottery of choice objects where each agent (simultaneously or iteratively) reports own preference and a sequence of finite length K preference profiles.

• For any small positive probability, e, arbitrarily close to 0, the mechanism selects dictatorial outcomes in the last messages each of which is implementable, and for each stage less than K the mechanism perturbs pure objective x in the original SCF mixed with another object, say, x-bar. If unanimous the origial is the case. Only if the cases of a single deviated agent who reports different profile from others, the same lottery will be selected and a small fine $(1/K) for the deviator. Otherwise x-bar to be selected. Other than the disagree fine, additional fine, $1, penalty for the last revision. Truthful equilibrium brings about the right outcome with probability, (1-Ne), i.e., almost 1.

• Abreu and Matsushima[1] proved the theorem by using above mechanism that assuming that for any pair of preference profile (w,w1) such that each agent has same preference under w and w1 iff the SCF has same value, the truth-telling in every stage is a unique iteratively undominated strategy message profiles (if N>2 or if N=2 and truth-telling profiles is NE). The assumption may be dropped if renegotiation is available[3].

On advantages and criticisms• This type of mechanism is very powerful and robust because it can implement any soc

ial choice rule approximately in iteratively undominated strategy equilibrium [1]. (Cf., Single stage implementation model requires that SCF must be monotonic given possible preference profiles.)

• By making use of Dekel-Fundenberg-Bogers procedure which permits an elimination of weakly dominated strategies only at the first step in the mechanism, it yields unique outcome which is order independent for the sequence of reports [2].

• Criticism for the A-M mechanism was firstly pointed out by Glazer and Rosenthal with response by Abreu and Matsushima in Econometrica 60, pp.1435-42, 1992. They make a case where the principal can implement a non-Pareto optimal outcome by using AM mechanism. Apparently, the problem is not that of suggesting vulnerableness of the AM mechanism.

• Since no longer the constraint on the set of implementable SCFs, as stated by one of the authors [4], it would be issued that the design of SCF itself with the focal set, i.e., which SCF should be implemented including that seems, at least at first glance, irrational.

• Virtual Bayesian implementation is incomplete information analogue of virtual implementation under complete information. It is questionable whether Bayesian monotonicity can be dispensed as well as Maskin monotonicity in the complete information environment[6]. Sufficient conditions has been argued by Abreu and Matsushima (A-M measurability), Duggan (incentive consistency), and Serrano and Vohra.

A variant of AM mechanism for two agent case[3]

• Each agent reports a sequence of own willing effort levels. (message process as negotiation)

• After message process the manager should decide a small bonus for agent 1 according to the last report from agent 1 which is a unit of known implementable (i.e., dictatorial) outcome.

• Rewards are scaled proportionally to the time interval the profile being reported.

• Fine for the final revision of the report during negotiation.• Fine agent 2 if disagree as a side-payment to agent 1.• These fines will vanish in the equilibrium ( iterated domin

ance, correctly ) which yields the truthful reports.

A prolog simulation of AM mechanism: game parameters

% the original final time is 400 in [3].

game(parameter,time_limit,manager,4).

game(parameter,auxiliary_reward2,manager,1).

game(parameter,penalty,last_revision,2.00).

game(parameter,penalty,last_disagree_for_worker2,1.00).

game(parameter,utility_scale,manager,0.01).

Initialization for the protocol

game(virtual( time(0), true_state(S), reports([[],[]]) ), B,C,D):- B=players([worker1,worker2]), C=acts([report(null),report(null)]), D=payoffs([0,0]), game(parameter,true_state,nature,S).

Recursive protocol of negotiationgame(virtual( time(T), true_state(S), reports([H1,H2]) ), B,C,D):- B = players([worker1,worker2]), C = acts([report(R1),report(R2)]), D = payoffs([Y1,Y2]), length([_|H1],T), game(parameter,time_limit,manager,K), (T > K -> !,fail; true), T0 is T-1, A1=virtual( time(T0), true_state(S), reports([HR1,HR2]) ), B1=players([worker1,worker2]), C1=acts([report(RR1),report(RR2)]), D1=payoffs([X1,X2]), game(A1,B1,C1,D1), (RR1=null ->[H1,H2]=[HR1,HR2] ;[H1,H2]=[[RR1|HR1],[RR2|HR2]] ), %(to be continued)

% (continued) game( agency(0), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([U1,U2]) ), game( virtual_closing( time(T), true_state(S), reports([H1,H2]), last_revised(_LRT,_Renagade1) ), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([W1,W2]) % fines and bonus ), game(parameter,utility_scale,manager,US), Y1 is X1 + US * U1/K - W1, Y2 is X2 + US * U2/K - W2. %nl,write(payoff(1):Y1 is X1 + US * U1/K - W1), %nl,write(payoff(2):Y2 is X2 + US * U2/K - W2).

Closing negotiationgame( virtual_closing( time(K), true_state(S), reports([H1,H2]), last_revised(LRT,Renagade) ), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([Fine1,Fine2]) ):- game(parameter,time_limit,manager,K), !, length([_|H1],K), game( penalty_for_last_revised(Renagade,LRT,[R1,R2],[H1,H2],K), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([FineA1,FineA2]) ), %penalty_for_last_disagree(worker2) game(parameter,penalty,last_disagree_for_worker2,PW2), (R1 =R2 -> FineB =0 ; FineB is PW2/K), game(parameter,auxiliary_reward2,manager,M), (R1 =S -> Bonus =M ; Bonus =0), Fine1 is FineA1 - FineB - Bonus, Fine2 is FineA2 + FineB.

game(virtual_closing(_,_,_,_),_,_,payoffs([0,0])).

Computing finesgame( penalty_for_last_revised(Renagade,LRT,[R1,R2],[H1,H2],K), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([FineA1,FineA2]) ):- min( LRT1,((nth1(LRT1,H1,RB1),RB1\=R1,!);LRT1 is K )), min( LRT2,((nth1(LRT2,H2,RB2),RB2\=R2,!);LRT2 is K )), ((LRT1=LRT2,LRT1>=K)->(Renagade=non,LRT=K,!);true), ((LRT1=LRT2,LRT1<K)->(Renagade=both,LRT=LRT1,!);true), (LRT1<LRT2->(Renagade=worker1,LRT=LRT1,!);true), (LRT1>LRT2->(Renagade=worker2,LRT=LRT2,!);true), %nl,write(last_revision:LRT-Renagade), game(parameter,penalty,last_revised,Penalty), L is Penalty / K, G = 0, %G is -L, member( [Renagade,FineA1,FineA2], [[worker1,L,G],[worker2,G,L],[both,L,L],[non,0,0]] % It will not wrok well if we specify the fine as follows analogically in Ref. 3, p.4, ll.12. %[[worker1,L,G],[worker2,G,L],[both,0,0],[non,0,0]] ).

A test run : the final state?- [am01].

-------------------------------------------------------------------% implementation problem in agency---- Abreu-Matsushima mechanism-------------------------------------------------------------------Warning: (d:/prolog/source/am01.pl:820): Clauses of game/4 are not together in the source-file% am01 compiled 0.09 sec, 10,748 bytes

Yes?- nash(virtual(time(4),true_state(success1),reports([[R,R,R],[R,R,R]])),P,A,U).

R = success1P = players([worker1, worker2])A = acts([report(success1), report(success1)])U = payoffs([2.8, 1.2]) ;

R = success2P = players([worker1, worker2])A = acts([report(success1), report(success1)])U = payoffs([1.85, 1.15]) ;

No?-

Simulation of the AM mechanism

game(virtual(S), players([worker1,worker2]), acts([report([R1|H1]),report([R2|H2])]), payoffs([Y1,Y2]) ):- game( parameter, time_limit, manager, K ), game( virtual( time(K), true_state(S), reports([H1,H2]) ), players([worker1,worker2]), acts([report(R1),report(R2)]), payoffs([Y1,Y2]) ).

% ---------------- test run ----------------

?- nash(virtual(S),P,acts([A,B]),U).

S = success1

P = players([worker1, worker2])

A = report([success1, success1, success1, success1])

B = report([success1, success1, success1, success1])

U = payoffs([2.8, 1.2])

S = success2

P = players([worker1, worker2])

A = report([success2, success2, success2, success2])

B = report([success2, success2, success2, success2])

U = payoffs([2.2, 1.8]) ;

No

Experimental lower bound of auxiliary reward

Other Parameters:

penalty

last_revision,2.

last_disagree,1.

utility_scale, 0.01.

?- nash(virtual(S),P,acts([A,B]),U).

S = success1P = players([worker1, worker2])A = report([success1, success1, success1, success1, success1])B = report([success1, success1, success1, success1, success1])U = payoffs([2.36, 1.2]) ;

S = success2P = players([worker1, worker2])A = report([success2, success2, success2, success2, success2])B = report([success2, success2, success2, success2, success2])U = payoffs([1.76, 1.8]) ;

No?- game(parameter,auxiliary_reward2,manager,Z).

Z = 0.56No?-

final time lower bound1 0.80012 1.30013 0.944 0.755 0.56

Modeling inductive playgame(virtual_inductive( time(K), true_state(S), reports([H1,H2]) ),B,C,D):- game(parameter,time_limit,manager,K), A=virtual( time(K), true_state(S), reports([H1,H2]) ), B=players([worker1,worker2]), C=acts([report(_R1),report(_R2)]), D=payoffs([_U1,_U2]), game(A,B,C,D).

game(virtual_inductive( time(T1), true_state(S), reports([H1,H2]) ),B1,C1,D):- A1=virtual( time(T1), true_state(S), reports([H1,H2]) ), B1=players([worker1,worker2]), C1=acts([report(R1),report(R2)]), D1 = payoffs([_U1,_U2]), game(parameter,time_limit,manager,K), length([_|L],K), nth1(T1,L,T1), length([_|H1],T1), T0 is T1 + 1, A = virtual_inductive( time(T0), true_state(S), reports([[R1|H1],[R2|H2]]) ), B = players([worker1,worker2]), C = acts([report(_RR1),report(_RR2)]), D = payoffs([_Y1,_Y2]), nash(A,B,C,D), game(A1,B1,C1,D1).

% alternative: in order to% utlize the experimental log% of the previous steps.

A = log_inductive1( T0, S, [[R1|H1],[R2|H2]] ),% another brief form for test run.

game(virtual_inductive(T,S,[H1,H2]),P,A,U):- game( virtual_inductive( time(T), true_state(S), reports([H1,H2]) ), P, A, U ).

% for the sake of the above.

game_ log (A1,B1,C1,D1),

Experimention of inductive play

?- game(test_inductive1(1,S1,H),P,A,U).

S1 = success1H = [[], []]P = players([worker1, worker2])A = acts([report(success1), report(success1)])U = payoffs([2.8, 1.2]) ;

S1 = success2H = [[], []]P = players([worker1, worker2])A = acts([report(success2), report(success2)])U = payoffs([2.2, 1.8]) ;

No?-

references• [1] D. Abreu and H. Matsushima (1992). Virtual implementation. Eco

nometrica 60: 993-1008.• [2] D. Abreu and H. Matsushima (1995). Exact implementation. Jour

nal of Economic Theory 64: 1-19.• [3] H. Matsushima (1996). Mekanizumu dezain no gehmu riron. [Ga

me theory of mechanism design.] Keizaigaku Ronshu 62(1): 2-12.• [4] H. Matsushima (1996). A-M mekanizumu no gohrisei. [Rationality

of A-M mechanism.] Keizai Kenkyu 47(1): 1-15.• [5] L.C. Corchon (1996). The Theory of Implementation of socially O

ptimal Decisions in Economics. Macmillan/St. Martin’s Press. • [6] R. Serrano and R. Vohra (2001). Some limitations of virtual Baye

sian implementation. Econometrica 69(3): 785-792.