Constrained Delegation

Lessons for Behavioral Mechanism Design

Munich

December 2025

Brandon Williams

Alistair Wilson

Richard Van Weelden

Setup

Tend to get inefficiency when we consider interactions between uninformed and informed parties:
- Sender-receiver
- Bargaining
- Market for lemons
In some of these settings, delegating decision power to the informed party:
- Potential for efficient solution
- But share of surplus created is smaller for the party ceding power

Motivation

Constrained Delegation allows for an intermediate outcome:

The uninformed party delegates some of their decision power to the informed party
But imposes sone constraints on the choices the informed party can make
- For example, minimum and maximum limits

Veto bargaining provides an economically important setting to examine this:

Veto represents an outside option
Theory identifies constrained delegation as optimal mechanism (Kartik, Kleiner, and Van Weelden, 2021)
Simple setting for a clear experimental assessment!

Setup

Consider the following bargaining context:

Nature determines the informed party's preference
An uninformed principal makes a proposal
An informed agent responds, comparing the offer it to her outside option

Findings

Comparing Constrained Delegation to Take-it-or-Leave it offers we find:
- Constrained delegation does increase efficiency a lot
- However, the informed party still captures almost all of the benefits behaviorally
- Proposers fail to optimize in setting the constraints, providing more latitude
  - This increases both parties' expected payoffs!

Literature

Models of similar bargaining forms have been proposed in theory...

Under complete information - Romer and Rosenthal (1978)
Under cheap talk negotiation - Matthews (1989)
With valuable expertise involved - Holmström (1977)
More recently: Kartik, Kleiner, Van Weelden (2021)

And used in experiments...

Early bargaining - e.g. Roth and Murnighan (1980, 1982)
Vetoes within committees - Kagel, Sung, and Winter (2010)
Multiple rounds of bargaining - Nunnari (2021)

More recently: Kartik, Kleiner, Van Weelden (2021)

Theory

Theory

Proposer

Theory

Proposer

Proposer has:
- Increasing payoff \( \pi (z) \) over the outcome \( z \)
  - for simplicity, we'll assume \(\pi(z)=z\)
- Will make an offer to the vetoer

Theory

Vetoer

Proposer

\( \theta \)

Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)

Theory

???

\( \theta \)

Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)

Theory

???

Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)
- A choice \( z \in \{0,Y\} \), either the veto threat point (here 0) or in the offer

\( \theta \)

Theory

Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)
- A choice \( z \in \{0,Y\} \), either the veto threat point (here 0) or in the offer

Theory: Take it or Leave It

Proposer

Offer \( y \)

\( \frac{ y}{2} \)

These types veto

These types take offer

\( z = 0 \)

\( z = y \)

Theory: Take it or Leave It

Proposer

Offer \( y \)

\( \frac{ y}{2} \)

Suboptimal deals:

inefficient as \( \theta > y \)

preferred by both

Breakdown:

inefficient as \( \theta > 0 \) preferred by both

Theory: TIOLI equilibrium

Proposer increasing the offer \(y\) leads to:
- Marginal gain of \(\pi'(y)\cdot(1-F(\tfrac{y}{2}))\)
- Marginal loss of \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)

\(\tfrac{y}{2}\)

\(z=0\)

\(z=y\)

\(y\)

\(f(\theta)\)

Theory: TIOLI equilibrium

\(\tfrac{y}{2}\)

Proposer increasing the offer \(y\) leads to:
- Marginal gain of \((1-F(\tfrac{y}{2}))\)
- Marginal loss of \(\tfrac{y}{2}\cdot f(\tfrac{y}{2})\)

\(z=0\)

\(z=y\)

\(y\)

\(f(\theta)\)

Theory: Delegation

\( \theta \)

Vetoer ideal

Proposer

Proposer:
- Offers set of options \(Y\)
Vetoer has:
- Chooses \(z\in\left\{0\right\}\cup Y\), either the veto threat point (here \(0\)) or some offer in delegation set

Theory: Delegation

\( \frac{ y}{2} \)

These types veto

These types

choose minimum offer

\( z = 0 \)

\( z = y \)

Offer \( [y,1] \)

\( y \)

These types

choose their ideal point

\( z = \theta \)

Theory: Delegation

\( \frac{ y}{2} \)

Offer \( [y,1] \)

\( y \)

No suboptimal deals:

The delegation mechanism ensures

these options are available

Breakdown:

inefficient as \( \theta > 0 \)

Theory: Delegation Equilibrium

Proposer increasing the minimal offer \(y\) leads to:
- Marginal gain of \(\pi'(y)\cdot(F(y)-F(\tfrac{y}{2}))\)
- Marginal loss of \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)

\(\tfrac{y}{2}\)

\(y\)

\(f(\theta)\)

Theory: Delegation Equilibrium

\(\tfrac{y}{2}\)

\(y\)

\(f(\theta)\)

Proposer increasing the offer \(y\) leads to:
- Marginal gain of \((F(y)-F(\tfrac{y}{2}))\)
- Marginal loss of \(\tfrac{y}{2}\cdot f(\tfrac{y}{2})\)

Theory: Summary

Delegation should substantially reduce inefficiency
- Remove all the suboptimality by providing choice
- By changing the margins, leads to more latitude in offers, so decreasing breakdown too
Delegation should improve the proposer's payoffs
Proposers offers should shift with the distribution of responders

Experiment

Experimental Design

Construct environment that directly follows the veto-bargaining framework
One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?

Experimental Design

Construct environment that directly follows the veto-bargaining framework
One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?

Experimental Design

Construct environment that directly follows the veto-bargaining framework
One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?

Experimental Design

Proposer

Seller

Vetoer

Buyer

State

Ideal Demand

Offer

Widgets

Delegation

Widget Menu

Types

Urn Draws

Delegation treatment:

offer a range

Take it or leave it:

single offer

Experimental Design

Within-subject variation:
- Varying distributions (high, middle, low) for the Buyer
- Changing roles: 5 rounds in one role, 5 rounds in the other, and back to first role for 5 more rounds
Between subject 2 x 2:

	No Chat	Chat
Take-it-or-leave-it	N=66	N=60
Delegation	N=64	N=66

Experimental Design

Within-subject variation:
- Varying distributions (high, middle, low) for the Buyer
- Between subject 2 x 2
Collect other behavioral variables (identification through subtraction):
- Robot Buyer: remove strategic uncertainty/other regarding
- Pure incentives: present mechanism choice as lottery (remove mechanism complexity)
- Pure allocation: remove uncertainty and mechanism

Results

Sanity check: Sellers respond to alignment

Low

Middle

High

Low

Middle

High

Sanity check: Sellers respond to alignment

Low

Middle

High

Minimal offer in interval:

Sanity check: Sellers respond to alignment

Low

Middle

High

Minimal offer in interval:

Sanity check: Sellers respond to alignment

Other quick results

Buyers overwhelmingly pick the best option available
- They pick the best number of widgets from the menu, and take the outside option when better
Sellers (mostly) keep offers higher than the minimum
- They mostly refrain from setting the upper bound

Sellers offer more latitude under delegation

Low

Middle

High

Low

Middle

High

Sellers offer more latitude under delegation

Delegation should increase efficiency

Take it or Leave It

Delegation

Delegation should increase efficiency

Take it or Leave It

Delegation

Take it or Leave It is inefficient

Take it or Leave It

Delegation

Inefficiency not eliminated under delegation

Take it or Leave It

Delegation

Delegation does increase efficiency

Take it or Leave It

Delegation

Take it or Leave It

Delegation

Delegation does increase efficiency

Who benefits from delegation?

Buyers mostly benefit from delegation

What explains behavioral deviations?

Main reason sellers don't extract more of the delegation gain is optimization failure:

Remove player: Robot choices closely match delegation choices

What explains behavioral deviations?

Remove player: Robot choices closely match delegation choices
Remove mechanism complexity: Lottery choices indicate pure preference is for less delegation

Other Choices

Replace delegation decisions with lottery?

Results: Key Points

Offers respond to the expected type in a well-ordered manner
- Sellers offer more latitude than theoretical prediction
- Sellers in delegation offer ever more latitude
Delegation mechanism is more efficient than take it or leave it
Most of the "cost" of the mechanism falls on the Seller:
- More of the efficiency gains are captured by the Buyer
- Best improvements for the Seller is when alignment is low
- Data suggest that the optimization failures are a reason for greater latitude in delegation
Addendum: pre-play communication is just as efficient as delegation

Communication Efficiency

Communication Offers

Conclusion

Examine constrained delegation
- Here in an ideal setting (veto bargaining)
Proposers respond to key tensions in the predicted directions
Clear efficiency gains from delegation
- But more of the surplus goes to the responder
Optimization failure in using the mechanism leads to over-delegation
- However, in expectation this error helps increase both their payoffs!