Theodore Groves: Incentives in Teams Casper Storm Hansen June 29, - PowerPoint PPT Presentation

Theodore Groves: Incentives in Teams Casper Storm Hansen June 29, 2009

General organization team model: T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] ( n + 1 ) -person game: G = [ I , ( S , S , P ) , { B i , i ∈ I } , { ω i , i ∈ I } ] Set of decision makers: I = { 0 , 1 , . . . , n } 1 The organisation head: 0 His employees: 1 , . . . , n Probability space of alternative states: ( S , S , P ) 2 State space: S σ -algebra over S : S (family of subsets of S that includes S and is closed under complementation and countable unions) Probabilty measure: P (countable additive function S → [ 0 ; 1 ] s.t. P ( ∅ ) = 0 and P ( S ) = 1) Set of alternative strategies for decision maker i : B i ( i ∈ I ) 3 → Set of joint strategies: B = � n i = 0 B i Payoff (compensation) function for decision maker i : 4 ω i : B × S → R (assumed to be P -integrable for every β ∈ B )

Expected value of the payoff function for decision maker i : ω i : B → R defined by ¯ � ω i ( β ) = ¯ ω i ( β, s ) d P ( s ) S A joint strategy β ∗ ∈ B is optimal if ω 0 ( β ∗ ) = max ¯ β ∈ B ¯ ω 0 ( β ) Assumption A: There exists a β ∗ ∈ B such that ω 0 ( β ∗ ) ≥ ω 0 ( β ) for all β ∈ B ω 0 ( β ∗ ) > ω 0 ( β ∗ /β i ) for all β i ∈ B i , β i � β ∗ ( i = 1 , . . . , n ) i For a joint strategy β = ( β 0 , . . . , β n ) and a strategy β ′ i for decision maker i , β/β ′ i is ( β 0 , . . . , β i − 1 , β ′ i , β i + 1 , . . . , β n )

Incentive structure: A set W = { ω i , i = 1 , . . . , n } of employee payoff functions. An incentive structure W ∗ = { ω ∗ i , i = 1 , . . . , n } is optimal if ω ∗ i ( β ∗ ) = max ω ∗ i ( β ∗ /β i ) ¯ β i ∈ B i ¯ uniquely for all i = 1 , . . . , n (the optimal joint strategy is in a strong sense a Nash equilibrium) The incentive problem: To find an optimal incentive structure.

✦ ✪ ✦ ✪ ✦ ✦ The paid worker incentive structure W 0 = ( ω 0 1 , . . . , ω 0 n ) is defined by if β i = β ∗ � 1 ω 0 i i ( β, s ) = ( i = 1 , . . . , n ) 0 otherwise

✦ ✪ ✦ ✪ ✦ ✦ The paid worker incentive structure W 0 = ( ω 0 1 , . . . , ω 0 n ) is defined by if β i = β ∗ � 1 ω 0 i i ( β, s ) = ( i = 1 , . . . , n ) 0 otherwise The profit-sharing incentive structure W I = ( ω I 1 , . . . , ω I n ) is defined by ω I i ( β, s ) = α i ω 0 ( β, s ) + A i ( i = 1 , . . . , n ) where α i is a positive constant and A i is any constant

The paid worker incentive structure W 0 = ( ω 0 1 , . . . , ω 0 n ) is defined by if β i = β ∗ � 1 ω 0 i i ( β, s ) = ( i = 1 , . . . , n ) 0 otherwise The profit-sharing incentive structure W I = ( ω I 1 , . . . , ω I n ) is defined by ω I i ( β, s ) = α i ω 0 ( β, s ) + A i ( i = 1 , . . . , n ) where α i is a positive constant and A i is any constant W 0 W I W II ✦ ✪ ✦ Compensation by individual performance ✪ ✦ ✦ Only requires limited knowledge of the head

General organization team model: T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] ( n + 1 ) -person game: G = [ I , ( S , S , P ) , { B i , i ∈ I } , { ω i , i ∈ I } ] Set of decision makers: I = { 0 , 1 , . . . , n } 1 The organisation head: 0 His employees: 1 , . . . , n Probability space of alternative states: ( S , S , P ) 2 State space: S σ -algebra over S : S (family of subsets of S that includes S and is closed under complementation and countable unions) Probabilty measure: P (countable additive function S → [ 0 ; 1 ] s.t. P ( ∅ ) = 0 and P ( S ) = 1) Set of alternative strategies for decision maker i : B i ( i ∈ I ) 3 → Set of joint strategies: B = � n i = 0 B i Payoff (compensation) function for decision maker i : 4 ω i : B × S → R (assumed to be P -integrable for every β ∈ B )

Condition S.1 (the decision makers) I = { 0 , 1 , . . . , n } , where i = 0 is the head and i = 1 , . . . , n the subunit managers.

Condition S.2 (independence of subunits) � � n � � n � , � n � ( S , S , P ) = , where ( S i , S i , P i ) is i = 0 S i , σ i = 0 S i i = 0 P i the probability space of the i th component’s environmental state � � n � variable and σ i = 0 S i is the σ -algebra of subsets of S generated by the σ -algebras S i , i = 0 , . . . , n

Condition S.3 (a strategy contains an observation strategy, a message strategy, and a decision strategy, and the subunit managers only communicate with the head) If β i ∈ B i then β i = ( ζ i , γ i , δ i ) for some observation strategy ζ i : S i → Y i message strategy γ i : Y i → Y i n except γ 0 : Y 0 → Y 0 of the form λ x γ 0 ( x ) = λ x ( γ 1 0 ( x ) , . . . , γ n 0 ( x )) ( γ i 0 : Y 0 → Y 0 and γ i 0 ( x ) is interpreted as the message from the head to the i th subunit) and decision strategy δ i : Y i → D i where Y 0 = Y 0 × · · · × Y n and Y i = Y i × Y 0 are information sets and D 0 , . . . , D n are decision sets. For given observation and message strategies, information functions y i : S → Y i satisfy � � ζ i ( s i ) , γ i y i ( s ) = 0 ( y 0 ( s )) ( i = 1 , . . . , n ) y 0 ( s ) = [ ζ 0 ( s 0 ) , γ 1 ( y 1 ( s )) , . . . , γ n ( y n ( s ))]

Condition S.4 (payoff for the head is the sum of the profits of the subunits and the central administration) The payoff function for the head is of the form n � ω 0 ( β, s ) = ν i [ δ i ( y i ( s )) , δ 0 ( y 0 ( s )) , s i ] + ν 0 [ δ 0 ( y 0 ( s )) , s 0 ] i = 1 where ν i : D i × D 0 × S i → R , i = 1 , . . . , n and ν 0 : D 0 × S 0 → R (profit functions)

Condition S.5 (The profit of a subunit accrues directly to that subunit) ω i ( β, s ) = ν i [ δ i ( y i ( s )) , δ 0 ( y 0 ( s )) , s i ] + . . . ???

The class I of all incentive structures requiring the head to know no more than y 0 ( s ) : The class of all tuples ( ω 1 , . . . , ω n ) where ω i ( β, s ) = ν i [ δ i ( y i ( s )) , δ 0 ( y 0 ( s )) , s i ] + C i ( y 0 ( s )) where again C i : Y 0 → R

Conditional expected value: For (measurable) subsets U ⊆ S : f ( s ) d P ( s ) � E [ f ( s ) | s ∈ U ] = P ( U ) s ∈ U � f ( s ) d ˆ = P U ( s ) s ∈ U � f ( s ) d ˆ = P ( s ) s ∈ U

The own profit incentive structure W II = ( ω II 1 , . . . , ω II n ) is defined by ω II i ( β, s ) = ν i [ δ i ( y i ( s )) , δ 0 ( y 0 ( s )) , s i ] + C II i ( y 0 ( s )) ( i = 1 , . . . , n ) where for all y 0 ∈ Y 0 � � � � d ˆ C II δ ∗ j ( y ∗ j ( s )) , δ ∗ 0 ( y ∗ i ( y 0 ) = 0 ( s )) , s j P ( s ) − A i ( i = 1 , . . . , n ) ν j { s ∈ S | y ∗ 0 ( s )= y 0 } j � i where again � � y ∗ ζ ∗ j ( s j ) , γ j ∗ 0 ( y ∗ j ( s ) = 0 ( s )) ( j = 1 , . . . , n ) y ∗ 0 ( s ) = [ ζ ∗ 0 ( s 0 ) , γ ∗ 1 ( y ∗ 1 ( s )) , . . . , γ ∗ n ( y ∗ n ( s ))] , and A i is any constant ( i = 1 , . . . , n )

Theorem Given the organization model T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] with the conglomerate specifications S.1-S.5, if T satisfies Assumption A, then W II is an optimal incentive structure in the class I .

Theorem Given the organization model T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] with the conglomerate specifications S.2-S.4, if T satisfies Assumption A and γ ∗ i [ Y i ] = Y i and ∀ y i ∈ Y i : P { s ∈ S | γ ∗ i ( y ∗ i ( s )) = y i } > 0 ( I = 1 , . . . , n ) , then W II is an optimal incentive structure in the class I .

Theorem Given the organization model T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] with the conglomerate specifications S.2-S.4, if T satisfies Assumption A and γ ∗ i [ Y i ] = Y i and ∀ y i ∈ Y i : P { s ∈ S | γ ∗ i ( y ∗ i ( s )) = y i } > 0 ( I = 1 , . . . , n ) , then W II is an optimal incentive structure in the class I . ω i ( β, s ) = ν i [ δ i ( y i ( s )) , δ 0 ( y 0 ( s )) , s i ] + C i ( y 0 ( s )) ω II i ( β, s ) = ν i [ δ i ( y i ( s )) , δ 0 ( y 0 ( s )) , s i ] + C II i ( y 0 ( s ))

Theorem Given the organization model T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] with the conglomerate specifications S.2-S.4, if T satisfies Assumption A and γ ∗ i [ Y i ] = Y i and ∀ y i ∈ Y i : P { s ∈ S | γ ∗ i ( y ∗ i ( s )) = y i } > 0 ( I = 1 , . . . , n ) , then W II is an optimal incentive structure in the class I . ω II i ( β ∗ ) = max ω II i ( β ∗ /β i ) uniquely for all i = 1 , . . . , n To be shown: ¯ β i ∈ B i ¯

Theorem Given the organization model T = [ I , ( S , S , P ) , { B i , i ∈ I } , ω 0 ] with the conglomerate specifications S.2-S.4, if T satisfies Assumption A and γ ∗ i [ Y i ] = Y i and ∀ y i ∈ Y i : P { s ∈ S | γ ∗ i ( y ∗ i ( s )) = y i } > 0 ( I = 1 , . . . , n ) , then W II is an optimal incentive structure in the class I . ω II i ( β ∗ ) = max ω II i ( β ∗ /β i ) uniquely for all i = 1 , . . . , n To be shown: ¯ β i ∈ B i ¯ ω 0 ( β ∗ ) = max ω 0 ( β ∗ /β i ) uniquely for all i = 1 , . . . , n Assumption A: ¯ β i ∈ B i ¯