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Contracting with Imperfect Commitment and the Revelation Principle: The Single Agent Case

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### Contracting with Imperfect Commitment and the Revelation Principle: The Single Agent Case

Helmut Bester & Roland Strausz

(As told by

Daniel Brown & Justin Tumlinson

30 January 2006)

Model Vocabulary I

T, set of agent types

- t T

, the probability distribution of the agent’s type

- t, the unconditional probability that the agent’s type is t

M, message set from which the agent may select

- m M
- principal selects the set M

X, set of decisions to which the principal can contractually commit

- x() X
- (M, x()), contract or mechanism
- x(m),agent can enforce x by sending message m

Y, set of decisions to which the principal cannot contractually commit himself

- y() Y
- y(m) means message received by the principal determines principal’s actions through model properties (e.g. optimality conditions), not that a message m commits the principal to a specific action per se

F(x(m)), correspondence restricting feasible choices in Y, given x(), m

- y(m) F(x(m))

Model Vocabulary II

Vt(x(m),y(m)), payoff of the principal if the agent is type t and sends m

Ut(x(m),y(m)), payoff of the agent if the agent is type t and sends m

Q, set of strategies from which a type t agent chooses a strategy, qt Q

- qt (m), the probability that a type t agent will send message m
- E.g. if |M| = 6, then qt = (0, ½, 0, 0, ⅛, ⅜) means a type t agent will send messages 2, 5 and 6 with probabilities ½, ⅛ and ⅜ respectively, a mixed strategy; qt (1) = 0, qt (2) = ½,… , qt (5) = ⅛, qt (6) = ⅜

p, the posterior belief vector of the principal’s belief about the agent’s type

- pt (m), the belief of the principal that the agent is type t, given m

V(q(),y(),x()|M) = E[payoff for principal]

= T t [M qt (m) Vt(x(m),y(m)) dm]

Ut(q(),y(),x()|M) = E[payoff for type-t agent]

= M qt (m) Ut(x(m),y(m)) dm

Model Timing

- Mechanism (M, x()) “induces” the principal-agent game
- Assumption: principal chooses M and x(), but X, Y, T, F(), V() & U() are exogenous
- Agent privately observes his type, t
- Agent chooses (mixed) strategy qt
- Message m sent according to strategy qt
- Principal updates beliefs, p, on agent’s type
- Principal chooses decision y F(x(m))
- Payoffs Vt(x(m),y(m)) & Ut(x(m),y(m)) realized

PBE Conditions

- Principal decides optimally given beliefs
- m M, y’ F(x(m))T pt(m) Vt(x(m),y(m)) T pt(m) Vt(x(m),y’(m))
- Agent anticipates y(m) & chooses payoff maximizing qt
- M qt(m) Ut(x(m),y(m)) dm M qt’(m) Ut(x(m),y(m)) dm, qt’ Q
- Principal’s posterior beliefs must be consistent with Bayes’ Rule
- pt(m) = (qt(m) t) / t’T (qt’(m) t’)

Incentive Conditions

- (q,p,y,x|M) is incentive feasible if (q,p,y) is a BPE given mechanism
- (q,p,y,x|M) is incentive efficient if incentive feasible AND… (q’,p’,y’,x’|M)
- V(q’,y’,x’|M) > V(q,y,x|M) and
- Ut(qt’,y’,x’|M) = Ut(qt,y,x|M) t T
- (q,p,y,x|M) and (q’,p’,y’,x’|M) are payoff equivalent if
- V(q’,y’,x’|M) = V(q,y,x|M) and
- Ut(qt’,y’,x’|M) = Ut(qt,y,x|M) t T
- Agent’s individual-rationality constraint:
- Ut(qt’,y’,x’|M) Ut0
- I.e. Principal must guarantee the agent can obtain his reservation payoff, Ut0, after the agent learns his type

Direct Mechanisms

- (M,x()) is direct if M = T
- Revelation Principle: Assume all decisions contractible. (q,x | M) incentive feasible a direct mechanism, d = (T,x’), and incentive feasible (q’,x’ | T) (q,x | M) and (q’,x’ | M) are payoff-equivalent. Moreover, qt(t) = 1 t T (i.e. the agent’s strategy is always truth-telling).
- Contracting problem reduces to

Maximize T t qt (t) Vt(x(t))

Subject to

Ut(x(t)) Ut(x(t’)) t’T incentive compatibility

Ut(x(t)) Ut0 t’T individual rationality

- But if the principal cannot commit to the entire allocation (x,y), the Revelation Principle is no longer applicable

Example

- In imperfect commitment setting, (M, x()), may support outcomes, that are not possible under a direct mechanism.
- Example (from paper): 2 types of agent, M={m1, m2, m3} Principal chooses agent’s speed s, and wage w. Can commit to w, but not s. Payoffs satisfy single crossing property:
- U1 = w-s2/5 U2 = w-ss/6 V1= 10s-s2-w V2= 10s-s2/4-wLet s(m1)=5, s(m2)=10, s(m3)=20

w(m1)=5, w(m2)=20, w(m3)=70

- It is possible to construct principals beliefs p so that they support s()
- An optimal strategy for the agent, is then mixed:q1= (3/4, 1/4, 0) q2= (0,1/2, 1/2)
- Precisely constructed so that p and q are consistent (Bayes’ Rule)

Example (Cont.)

- So (q,p,s,w | M) is incentive feasible but we don’t know about incentive efficient (it’s not).
- There is a positive probability that each of the three chosen speeds s is implemented. This is not possible under a direct mechanism T={t1,t2}!
- Why? Principal’s payoffs are strictly concave in s, at most 2 different speeds could be supported in PBE.
- Conjecture: For every m M, there exists a distinct y Y that is supported in a PBE?
- Note that the choice of w, placed no restrictions on choice of s.

Some Comments

- Connection between |M| and supportable y (s in example). This is in part due to the generality of the PBE conditions. There are lots of Equilibria.
- Principal has flexibility: M can be any metric space. Also, can choose beliefs that will support a choice of y and x(M).
- If we focus on incentive efficiency rather than incentive feasibility, we get tighter restrictions.
- In example, there is redundancy in the message space. Consider message’s effect on s. If mixture 1/3m3 + 2/3m1, then in expectation you get s=10, which is the same s, as if you sent m2

Messages are a means to distinguish types. Ideal situation for principal: Choose M such that each mi M is associated with a type ti

Proposition 1 Implications

- Support of q’ contains at most |T| messages
- p and q’ are consistent with Bayesian Updating
- Since replacing q with q’ does not change principal’s belief, y() remains optimal
- Principal’s expected payoff unchanged
- q’ is optimal strategy for agent
- Same expected payoff for agent using q or q’
- Agent indifferent between all messages selected with positive probability

A New Revelation Principle

- Linear independence of q’(mh) allows us the apply the marriage theorem. There exists a one to one mapping from M’ into T.
- Proposition 2: If (q,p,y,x| M) is incentive efficient, then their exists a direct mechanism d = (T, x*) and an incentive feasible (q*,p*,y*,x*| T) such that (q*,p*,y*,x*| T) and (q,p,y,x| M) are payoff equivalent. Moreover, qi*(ti)> 0 for all ti T.
- Differences compared to Revelation Principal? This only applies to incentive efficient allocations (rather than incentive feasible) and the agent doesn’t reveal his type with certainty. Rather revealing his true type is an optimal strategy which he chooses with positive probability.

Optimal Contracts

- With Proposition 2, we can formulate the principal’s problem as a standard programming problem with z =(x,y).
- Maximize tT t’M tqt(t’)Vt(z(t’))subject to
- Ut(z(t)) Ut(z(t’)) t’T incentive compatibility
- Ut(z(t)) Ut0 tT individual rationality
- [Ut(z(t))-Ut(z(t’))]qt(t’)=0 t, t’T
- y(t)argmaxyF(x(t)) t’T pt’(t)Vt’(x(t),y) tT optimality
- pt(t’) = (qt(t’) t) / t’’T (qt’’(t’) t’’) t, t’T Bayesian consistency

Some Comments

- Constraint 3: If it is optimal to the principal to induce the agent to misrepresent his type, the agent must be made indifferent between reporting the different types.
- It is possible that in the solution there is positive probability that an agent misrepresents his type.
- Finding Optimal Contract becomes computational problem
- Difficulty lies in determining which constraints are binding at the optimum.

Apply program to example: Constraint 4 implies principal chooses s(ti) to maximize p1(ti)V1(w,s)+ (1- p1(ti))V2(w,s)

Implies: s(ti)= 20/(1+3 p1(ti))

Constraint 3 is binding for t1 which implies q2(t1)=0, q2(t2)=1 and

U1(w(t2), s(t2)) = U1(w(t1), s(t1)) = 0

Check to see that other constraints are satisfied, use 5 to determine beliefs then do principal’s maximization problem.

Solution: s(t1)=5, s(t2)=10, w(t1)=5, w(t2)=20

Applications

- Multi Stage contracting
- Contract binds for first period, then gets renegotiated in the next period. In 2nd period, agent may insist on original contract. So first period contract x imposes restriction on 2nd period choice y (the assumption y(m) F(x(m)) can be used).
- For more periods, consider that at each period, principal can commit to decision in that period, but not to ones in future periods. Future periods are discounted.
- Use Dynamic Programming techniques to solve.
- In solution, agents gradually reveal information of their type.

Circumventing the Problem (Techincals)

- Direct mechanisms may not support some indirect mechanisms
- Focus on efficient allocation
- Lemma 1: Let (q,p,y,x | M) be incentive efficient. Then |T| T pt(m) Vt(x(m),y(m)) = T pt(m) t / t
- Proposition 1: Let (q,p,y,x | M) be incentive efficient. Then incentive feasible (q’,p,y,x | M) and M’ with |M’| |T| and T,M’ tqt(m) = 1 (q,p,y,x | M) and (q’,p,y,x | M) are payoff equivalent. Moreover, vectors q’(m) = [q1(m), …, qt(m), …, q|T|(m)] are linearly independent m M’.

Constructing Payoff Equivalent q’

- Assume (q,p,y,x| M) is incentive efficient. Lemma 1 is FOC implied by assumption.
- Proposition 1: gives us q’ such that (q’,p,y,x| M) is payoff equivalent to (q,p,y,x| M), q’s support (M’) contains at most |T| elements, The set of vectors{q’(mh)} h=1,.., |M’| is linearly independent.
- Rescales q in a clever way (getting rid of redundancies in M) so that optimality condition, Bayes Rule are satisfied and principals beliefs don’t change (which then implies same choice of y)
- Why linear independence is important? we have almost gotten to the point where each element of M is associated with a specific type (think of changing to an orthogonal basis)

-This implies the principal can solve the contracting problem with a Message set of dimension |T|

Proposition 1 Technical Intuition

- |M| > |T| vectors q(m) = [q1(m),…, q|T|(m)] are linearly dependent
- From Bayes’ Rule: pt(m) = (qt(m) t) / t’T (qt’(m) t’)pt(m) q(m) = (qt(m) t), where q(m) = Prob{message = m} = t’T qt’(m) t’mM pt(m) q(m) = mM qt(m) tmM pt(m) q(m) = tmM p(m) q(m) =
- Thus P = conv({[p1(m),…, p|T|(m)] : mM })
- dim(P) |T|-1 because tT pt(m) = 1
- Carathedory’s Theorem: Given a set S, for any point p in conv(S) there is a subset T with p in conv(T), with |T| = dim(S)+1, and the points of S' are affinely independent
- Carathedory’s Theorem mM p(m) (m) = has a non-negative solution at most |T| scalars (m) > 0
- Define qt’(m) = (m) qt(m) / q(m)mM qt’(m) = mM (m) qt(m) / q(m) = mM (m) pt(m) / mM t qt(m) = t / mM t qt(m) = 1 / mM qt(m) = 1 (i.e. it is a valid strategy)

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