Principal-Agent Games - Equilibria under Asymmetric Information -

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1 Principal-Agent Games - Equilibria under Asymmetric Information - Ulrich Horst 1 Humboldt-Universität zu Berlin Department of Mathematics and School of Business and Economics Work in progress - Comments welcome 1 Based on ongoing work with Francoise Forges and Robert Simon

2 Outline Motivating example Repeated games of incomplete information Static games The game (setup, examples,...) Approachability and individual rationality Joint-plan equilibria The mathematical core Contracts and cooperative games The general model Existence of equilibria compact type spaces finite type spaces Example: Risk sharing under asymmetric information

3 Motivating Example

4 Motivating Example Informed player (agent) with random endowment H wants to share his risk exposure with uninformed player (principal). Randomness is described by a probability space ˆ :=K, ˆ F := K F, P := p0 P where we assume for the moment that K < 1. For X : ˆ! R the agent s type-dependent utility is U k (X )=E k ( ) P[u(X )] while the principal s utility function is Û(X )=E p0 P[û(X )]

5 Motivating Example The agent sells contingent claims from a set X to the principal. A contract comprieses a stochastic kernel µ : K ) S and a mapping q : S! X where S is a space of signals. The principals accepts a claim q(s) 2 X only if E ps P[û(q(s))] 0 The agent has a type-dependent reservation utility; 8k 2 K: E k ( ) P[u(H + q(s))]µ(k; ds)) E k ( ) P[u(H)] Does a reasonable, optimal,... contract (µ, q) exist and what is the general mathematical structure behind this problem?

6 Part I

7 Static games

8 Static Non-Cooperative Games A 2-person (non-cooperative) 1-stage game is a described by: 1 := (I, J, A, B) where I and J are the are finite action spaces and the matrices A i,j, B i,j are the players payo functions.

9 Static Non-Cooperative Games The mixed-extension of the game 1 is defined as 1 := ( (I ), (J), h 1, h 2 ) where (M) is the set of probability distributions on the set M and the payo functions are: h 1 (s, t) := X s i A i,j t j =< s, At > i2i,j2j h 2 (s, t) := X i2i,j2j Notice that payo functions are bi-linear. s i B i,j t j =< s, Bt >.

10 Definition An equilibrium in mixed strategies of the game strategies (s, t )suchthat 1 is any pair of < s, At >= max s2 (I ) < s, At > and < s, Bt >= max t2 (J) < s, Btt >

11 Definition (Zero-Sum Games) The game 1 := (I, J, A, B) iscalledazero-sum game if B = A. Its value is given by w := max min s t < s, At >= minmax < s, At >. t s Remark The value of a zero-sum game and hence an equilibrium exists. The min-max approach only works for games with scalar payo s, not vector payo s.

12 Repeated games of incomplete information

13 Repeated Games Repeated game of incomplete information on one side ( 1 (p)): there are two players, labelled PI and PII there is a finite set K of types; nature choses k 2 K according to a known probability distribution p 2 (K) PI is informed about k 2 K, PII is not the payo matrices in state k 2 K are A k,respectivelyb k the game with payo matrices (A k, B k )isrepeatedinfinitely often after each state of play, both players are informed about each other s move payo s in the infinitely repeated game are the average payo s of the stage games.

14 Theorem (Zero-sum games) Consider a zero-sum (B k = A k ) repeated game of incomplete information and let 1(p) the 1-stage game with payo matrix A(p) := X k p k A k. The game 1 (p) has a value a (p). The value of the infinitely repeated zero-sum game 1 (p) is v 1 (p) :=Cav(a )(p) where Cav(u) is the concave envelope of u, i.e. the smallest concave function dominating u.

15 Remark It is clear, the PI can guarantee himself Cav(a )(p) in 1 (p). It less obvious that PII can prevent him from getting more. The argument uses Blackwell s approachability theorem, an extension of v. Neumann s min-max Theorem to games with vector-valued payo s.

16 Approachability

17 Definition Let C be a payo matrix, {(i t, j t )} be a sequence of moves and put c n := 1 n nx C it,jt. t=1 The sequence of random variables {c t } approaches asetc if d(c n, C )! 0 a.s. A set of (vector) payo s C is approachable for some player if he/she has a strategy such that d(c n, C )! 0 a.s. regardless of his/her opponent s strategy.

18 Theorem (Blackwell (1956)) In a repeated game with payo matrices A k for PI, a closed convex set of (vector) payo s C is approachable for PII if and only if a (q) apple max (q c) for all q 2 (K). (1) c2c Remark Consider an infinitely repeated game with payo matrices A k for PI and B k for PII. PII can force PI s payo to belong to a set C if (1) holds. PII can guarantee himself Vex(b (p)).

19 Individual Rationality A key concept in repeated games is (interim) individual rationality. Both players consider their respective zero-sums games: A(p) and B(p) with respective values a (p) andb (p). Using the arguments from the zero-sum case one sees that: PII can guarantee herself Vex(b (p)) in 1 (p) PII can prevent PI from getting more than x k (k 2 K) inthe same game if x q a (q) for all q 2 (K)

20 Definition (Individual rationality) A payo vector y 2 R K is called individually rational for PI if y q a (q) for all q 2 (K). A payo z 2 R is called individually rational for PII if z Vex(b (p)) for all possible p 2 (K).

21 Joint plans

22 Joint-Plan Equilibria A joint-plan is a cooperative agreement between the players; the main idea is as follows first change the uninformed player s belief about the distribution of types using signals, then play a fixed (joint) action for the rest of the game, punish the opponent if he/she does not follow the pre-agreed plan. More formally, a joint-plan comprises: a finite set of signals S and a signaling kernel µ : K ) S for any posterior probability p s a joint action x s 2 (I J) punishment strategies to be implemented when the opponent does not follow x s

23 Definition (Joint-plan equilibria) A joint-plan describes an equilibrium if there exists y 2 R K s.t. Incentive compatibility (IC) for PI: if µ(k; s) > 0, then y k =max s 0 Individual rationality (IR) for PI: X i,j A k i,jx s0 i,j y q a (q) 8q 2 (K) Acceptability (A) for PII: if µ(k; s) > 0 for some k 2 K, then X B i,j (p s )xi,j s Vex(b )(p s ) i,j

24 Theorem (Sorin ( 83); Simon, Spiez &Torunczyk ( 95)) A joint plan equilibrium exists Remark Sorin (1983) established existence of equilibrium for K =2; Simon et al. (1995) extended his result to general finite type spaces using methods and techniques from algebraic topology.

25 Amoreabstractpointofview A joint-plan equilibrium can viewed as a special case of a cooperative game where one party, the agent, has private information and one, the principal, has not the principal suggests a cooperative agreement (contract) to the agent as part of the agreement, information may be disclosed there are side-constraints (outside options) One looks for a cooperative agreement in equilibrium. But what is the participation constraint?

26 Some Comments In a repeated games framework the concept of individual rationality is based on the idea of approachability, a dynamic concept and the willingness to cooperate is observable (punishment)

27 Part II

28 Preferences and Types There is a finite (or compact) set K of agent-types There is a compact set X 1 of actions for the agent and a compact set of actions X 2 for the principal The preferences are described by concave utility functions U k : X 1 X 2! R The preferences of the principal are (not yet) important.

29 Interim Individual Rationality The agent can accept the principal s proposal or not. Upon acceptance, a signal is send, an action implemented Upon rejection, the following (1-stage) game is played: U k ( ) forallk 2 K and (X 1, X 2 ) A payo vector v =(v k )isinterimindividuallyrational if: 9x 2 2 X 2 s.t. 8(x 1, k) 2 X 1 K : U k (x 1, x 2 ) apple v k In the complete information case, this is just min max.

30 Lemma (Non-standard (IR) condition) Assume that the following function is convex for k =1, 2: and define a mapping f : k (x 2 )= max x 1 2X 1 U k (x 1, x 2 ) f (q) := min (K)! Rvia x 2 2X 2 X k q k k (x 2 ) Then, a payo vector v is interim individually rational if and only if q v f (q) for all q 2 (K).

31 Example (Cournot game with private information) Consider a type dependent utility function of the form U k (x 1, x 2 )=x 1 (d x 1 x 2 ) c k x 1 A direct calculation shows that that k : X 2! R is convex. Remark The individual rationality condition q v f (q) for all q 2 (K) now appears in a static game with concave utility functions. When f is convex, it reduces to v k f (k), that is, to a model with type-dependent outside options.

32 Equilibria in principal-agent models

33 A More Abstract Point of View A joint-plan equilibrium is a special case of the following model: there are two players: a principal and an agent compact type space K; distribution p 0 2 (K). agents have type-dependent concave utility function U k ( ) :X! R on a convex space of joint actions X a signaling kernel µ : K ) S akernelq : S ) X that choses joint actions Apair(µ, Q) willbecalledacontract.

34 A More Abstract Point of View: (IC) Incentive compatibility for the agent prevails if each type is indi erent between the results of his actions (not signals!): U k ( ) u k (µq)(k; )-a.s. and U k ( ) apple u k (p 0 µq)-a.s. In models of expected utility, this reduces to the condition that types are indi erent between the signals sent.

35 A More Abstract Point of View: (IR) Individual rationality in the game is described for the principal in terms of a non-empty correspondence F : (K) ) X for the agent in terms of a function f : (K)! R.

36 Definition (Equilibrium) A contract (µ, Q) isanequilibrium if the following conditions hold: (A) for the principal: for almost all (x, s) 2 X S: x 2 F (p x,s ) (IC) for the agent: for almost all types: U k ( ) = u(k) (µq)(k; )-a.s; U k ( ) apple u(k) (p 0 µq)-a.s. (IR) for the agent: for p 0 -a.e. k 2 K: u p f (p) forallp2 (K)

37 Theorem (Existence of equilibrium - compact type spaces) Suppose that the following conditions hold: X, S and K and compact metric spaces; K is Polish the utility function (k, x) 7! U k (x) is continuous for all k 2 Kthemapx7! U k (x) is concave the correspondence F : (K) ) X is u.s.c., non-empty, convex-valued and inversely convex: x 2 F (p) for all p 2 P, then x 2 F (q) 2 Conv(P) the function f : K! R is weakly continuous. Then an equilibrium exists (there is no reason to assume that Q is degenerate).

38 Theorem (Existence of equilibrium - finite type spaces) Suppose that K is finite and that the following conditions hold: X is a compact metric space for all k 2 Kthemapx7! U k (x) is concave the correspondence F : (K) ) X is u.s.c., non-empty, convex-valued. Then an equilibrium exists with a degenerate signaling kernel Q(s; dx) = x(s) (dx). In particular, no further randomization over actions is needed.

39 Application: risk sharing under asymmetric information

40 Back to the Motivating Example Chose the action space to be a set of contingent claims: mx X := { i=1 ix i : i 0, X i i =1} The correspondence F is given by F (p) :={X 2 X : E p P [û(x ))] 0} The function f is given by f (k) :=E k ( ) P[u(H)] Theorem If the type space is compact, then an equilibrium exists.

41 Preferences of Non-Expected Utility Type Same framework as above with K < 1, < 1 The agent s utility function is U k (X )= min Q2Q E k( ) Q[ X ] (Q) The principal has the utility function: n o U(X )= min E ˆQ[ X ] ˆ ( ˆQ). ˆQ2 Q ˆ p Furthermore: F (p) ={X : U(X ) 0}, f (k) =0. Theorem Under mild conditions on the map p 7! ˆ Q p,anequilibriumexists.

42 (A Conjecture on) Optimality Suppose principal can chose µ and has a preference functional G : (K)! R such as Z G( ) = {c(y) } (d(y, )) Then her objective would be max{g(p 0 µq) :(µ, Q) isanequilibrium} For one-dimensional types this is a standard problem addressed by convex analysis methods Theorem (Conjectured) Under mild conditions an optimal contract for the principal exists.

43 Conclusion Proved a general existence of equilibrium result for principal-agent models The method allows for multi-dimensional agent-types Application to risk-sharing under asymmetric information Future research: optimal contract design for multi-dimensional types partial revelation of information and dynamic models (zero-sum) repeated-games with incomplete information with uncountably many types...

44 Thanks!