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Shirking and Squandering in Sharing Games

Dennis Courtney, University of California, Berkeley
Thomas Marschak, University of California, Berkeley

A BEJTE Topics article.

Abstract

In a sharing game the players' choices yield a revenue, each player's choice carries a cost, and a player's payoff is a portion of the revenue minus the player's cost. Such games are appealingly simple devices for partially aligning individual incentives with organizational goals, but their equilibria may be inefficient, i.e., at an equilibrium the surplus (revenue minus the sum of the costs) may not be maximal. Sharing games as a general class have not been well studied. We start a general theory of sharing games by going beyond the common economic setting, where strategy sets are continua and cost and revenue functions are smooth. We include games in which some of a player's strategies are equally costly, and revenue changes when he switches from one of them to another. We consider several large classes of reward functions, including nondecreasing residual (NDR) functions, in which residual (revenue minus rewards) does not drop when revenue increases. That class includes budget-balancing functions, where residual is always zero. To focus the discussion, we examine a ``Folk Claim", which asserts that at every inefficient equilibrium shirking, in some sense, occurs. We show that in NDR games a complementarity condition indeed insures that no one squanders at equilibrium (spends more than at an efficient profile). But when we drop complementarity, the situation changes sharply, and there are games with compelling equilibria, at which some players squander. The shirking/squandering distinction is particularly important in tracing the effect of technical improvement on the surplus shortfall at a sharing game's equilibrium. The paper also obtains conditions for existence of (pure-strategy) equilibria and finds, in particular, that every finite game in which rewards are linearly related has an equilibrium.

Errata

Page 3 - Delete the third sentence ('Suppose...') in Footnote 4.

Page 11, Proof of Theorem 6 - The first two sentences should be replaced by the following three sentences:

For any subset V⊆N, let ρV ( ) denote Σ_i∈V ρ_i( ). Suppose,
contrary to our claim, that at x no player set W_t collectively shirks
relative to y. Then we have

τ(x_Wt) ≥ τ(y_Wt), τ(x_ŵt) ≥ τ(y_ŵt) for all t ∈{1,...,T},

where Ŵt is the complement of W_t.

Statement (i) should be:

ρ_Wt(Α(y_Wt, x_ŵt))≥ ρ_ŵt(Α(y)); ρ_ŵt(Α(y_ŵt, x_ŵt)) ≥ ρ_Wt(Α(y)).

Page 12, Proof of Theorem 6 - The last three sentences should be replaced by the following:

Adding (ii) and (iii) we obtain:

(iv) ρN(Α(x)) − τ(x) ≥ ρN(Α(y)) − τ(y)

Since residual is nondecreasing and since aggregative productivity implies Α(x) ≥ Α(y), we have

(v) Α(x) − ρN(Α(x)) ≥ Α(y) − ρN(Α(y))

Adding (iv) and (v), we obtain

Α(x) − τ(x) ≥ Α(y) − τ(y).

That contradicts our assumption that x is not efficient.

Page 26, Proof of Theorem 5 - Lines 3–5 should be replaced by the following:

...and any profiles w, x, y, z, the following holds:

If c_i (z_i) ≥ c_i (w_i) for all i ∈ Τ, and c_i (x_i) ≥ c_i (y_i) for all i ∈ −Τ, then
Α (z_Τ, x_−Τ) − Α(w_Τ, x_−Τ) ≥ Α (z_Τ, y_−Τ) − Α(w_Τ, y_−Τ)

Submitted: May 13, 2006 · Accepted: September 25, 2006 · Published: December 6, 2006

Originally published in Topics in Theoretical Economics.

Recommended Citation

Courtney, Dennis and Marschak, Thomas (2006) "Shirking and Squandering in Sharing Games," Topics in Theoretical Economics: Vol. 6 : Iss. 1, Article 21.
Available at: http://www.bepress.com/bejte/topics/vol6/iss1/art21