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
Theory
0
1
Theory
Proposer
0
1
Theory
Proposer
0
1
- 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
- Increasing payoff \( \pi (z) \) over the outcome \( z \)
Theory
Vetoer
0
1
Proposer
\( \theta \)
- Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)
Theory
???
0
1
\( \theta \)
- Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)
Theory
???
0
1
- 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
0
1
Theory: Take it or Leave It
0
1
Proposer
Offer \( y \)
\( \frac{ y}{2} \)
These types veto
These types take offer
\( z = 0 \)
\( z = y \)
Theory: Take it or Leave It
0
1
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 \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)
0
1
\(\tfrac{y}{2}\)
\(z=0\)
\(z=y\)
\(y\)
\(f(\theta)\)
Theory: TIOLI equilibrium
0
1
\(\tfrac{y}{2}\)
- 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
0
\( \theta \)
1
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
0
1
\( \frac{ y}{2} \)
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
0
1
\( \frac{ y}{2} \)
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 \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)
0
1
\(\tfrac{y}{2}\)
\(y\)
\(f(\theta)\)
Theory: Delegation Equilibrium
0
1
\(\tfrac{y}{2}\)
\(y\)
\(f(\theta)\)
- 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
Sanity check: Sellers respond to alignment
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?
Who benefits from delegation?
Buyers mostly benefit from delegation
What explains behavioral deviations?
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
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
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!
Thank you! 🙏
Questions or Comments?
Diagnosing the Failures: Pure Optimization
Diagnosing the Failures: Lotteries
Diagnosing the Failures: Distribution
Results: Inefficiency (Data with No Comm)
TIOLI
Delegation
Results: Inefficiency (Data with Comm)
TIOLI
Delegation
Results: Communication Offers (TIOLI)
Low
Middle
High
Chat
No Chat
Results: Offers (Delegation)
Low
Middle
High
Chat
No Chat
Delegation (BEDI)
By Alistair Wilson
