``if you can, you must.'' the evolutionary foundation of ... kubitz.pdf · evolutionary...

“If you can, you must.” The Evolutionary Foundationof Reference Point Choice and Loss Aversion

Greg Kubitz and Lionel Page

28 June 2019

Introduction Model Equilibrium Results Conclusion

Background

Evolutionary basis for utility functions

Hedonic utility incentivizes actions to maximize fitness

Robson (JPE 2001), Samuelson and Swinkels (TE 2006), Rayo andBecker (JPE 2007), Netzer (AER 2009)

Framework of principal (nature/evolution) and agent

Nature designs a utility function over outputs

Agent chooses action that leads to higher utility inexpectation (when distinguishable)

Kubitz (QUT) + Page (UTS) Reference Point Choice and Loss Aversion


Motivation

Literature gives evolutionary foundation for utility functions withreference dependence which can adapt to fit the agentspossibilities/opportunities

Utility function S-shaped: bounded with an inflection point.

Explanation for two aspects of Prospect Theory

This project: We investigate how loss aversion can emerge from anevolutionary process of providing incentives for agents to maximizetheir fitness.



Optimal Utility Functions

Nature faces biological constraints

Utility is bounded

Limited instruments to distinguish actions

Information about individual decisions are not specifically encodedin utility function

Detailed information observed by agent prior to making adecision is not used in designing the utility function

Limitations due to complexity of causal impact

Benefit of adaptability

Utility is steeper over decisions that are more likely to be relevant!



Rayo Becker (2007)

Nature and agent have symmetric information about decision

Output is a noisy function of action

Maximize efficiency of action using a step function

Idea: To maximize the efficiency of action choice, naturesprovides the maximal differential in utility function at thepoint where likelihood of more efficient action becomes larger



Rayo Becker (2007) - appendix

Agent has more information about decision than Nature

Use a series of differentials - size of jumps depend on thelikelihood of agent facing each decision.

If prior distribution is single peaked then optimal valuefunction is approximately S-shaped

y1 y2 y3

Vmax

V ∗(y)



This paper

Agent has information about prior distribution of potentialdecisions that Nature does not have

Information would be useful for Nature to better provide morefocused incentives

Nature designs a menu of value functions

We allow for agents to select their value function

Incentive compatibility - agents with a higher prior wants toselect value function designed for them

Two cases: Either agent has additional information choice prior toselecting efficiency of action or additional information is realizedafter efficiency is chosen.



Players and actions

Nature and a single agent

Nature offers a menu of value functions V to maximize theexpected output choice, y , of the agent.

Agent chooses a value function V ∈ V, and action, ϕ, tomaximize value

Cannot distinguish actions that lead to similar expected valuesAction ϕ ∈ [0, 1] a measure of efficiency

The output of the agent is a function of the state, s, andaction: y(s, ϕ).

y(s, ϕ) =

s with probability ϕ

s − c with probability 1− ϕ



Information

The state is comprised of two additively separable components:s = ω + z .

Agent observed state: ω ∈ Ω,Ω′, with Pr(ω = Ω) = p.

Choice specific information: z ∼ Fz , independent of ω, singlepeaked and symmetric around 0.

Distribution of state is commonly known



Timeline:

1. Nature offers menu of value functions, V.

2. Agent privately learns ω.

3. Agent chooses value function, V ∈ V.

4. Information about choice, z , is realized and agent choosesaction ϕ.

Two cases:

4a Agent chooses efficiency of action prior to choice informationis realized

4b Information about choice realized prior to agen’t efficiencychoice



Biological constraints

Constraint on Nature

Value functions in menu are uniformly bounded:V (y) ∈ [Vmin,Vmax].

We normalize bounds so that V (y) ∈ [0, 1].

Constraint on agent

Agent uniformly selects action ϕ ∈ [ϕmin, 1] where

E[V (y)|ϕmin, s] = E[V (y)|ϕ = 1, s]− ε.

and s is information known by the agent prior to decision ofefficiency.



Agent’s Problem

Given state s and chosen value function, Vω, lower bound onaction is

ϕmin(Vω, s) = 1− ε

E[Vω(s)− Vω(s − c)].

Given private information, ω, choice of value function maximizes

E [Vω|ω] =

∫z

∫ 1

ϕmin(·)Vω(y) Pr(y |ϕ(Vω, s), s) dϕ dF (z)



Agent’s Problem

Plugging in the value of ϕmin(V ω, s), the expected value forchosen value function ω

E [Vω|ω] =

∫zVω(ω + z)dF (z)− ε

2

It is sufficient for Nature to offer V = VΩ,VΩ′ which satisfy thefollowing IC constraints:∫

zVΩ′(Ω′ + z)dF (z) ≥

∫zVΩ(Ω′ + z)dF (z)∫

zVΩ(Ω + z)dF (z) ≥

∫zVΩ′(Ω + z)dF (z)



Nature’s Problem

Nature maximizes the expected efficiency of the agent’s action

maxVΩ,VΩ′

p

∫z

ϕmin(VΩ, s)f (z)dz + (1− p)

∫z

ϕmin(VΩ′ , s)f (z)dz

where

1 each value function is weakly monotonic,

2 both value functions are bounded by [0, 1] and

3 the IC constraints of the agent are satisfied.



Finite states and outputs

Consider the following distributions of states

ω ∈ Ω,Ω + 1, Pr(ω = Ω) = p

z ∈ −1, 0, 1 and f (z) =

1/2, z = 01/4, z = −1, 1

Output function with c = 1

y(ω, z , ϕ) =

ω + z with probability ϕ

ω − 1 + z with probability 1− ϕ

Five possible outputs: Ω− 2,Ω− 1,Ω,Ω + 1,Ω + 2.



Decision made prior to realization of z

Nature’s objective function

maxVΩ,VΩ+1

p

∫z

ϕmin(VΩ,Ω)f (z)dz + (1− p)

∫z

ϕmin(VΩ+1,Ω + 1)f (z)dz

Despite five possible outputs, only two efficiency choices need tobe incentivized.

Selection of value function informs Nature how to bestincentivize the agent

Optimal value functions are step functions for each type ofagent at modal decision



Ω− 2 Ω− 1 Ω Ω + 1 Ω + 2

1VΩ+1

VΩ

V (y)

Figure: Optimal value functions when p = 1/2.



Ω− 2 Ω− 1 Ω Ω + 1 Ω + 2

1VΩ+1

VΩ

V (y)

Figure: Optimal value functions when p ≈ 0.61.



Agent knows z prior to decision

Nature’s objective function

maxVΩ,VΩ+1

p

∫z

ϕmin(VΩ,Ω+z)f (z)dz+(1−p)

∫z

ϕmin(VΩ+1,Ω+1+z)f (z)dz

Each value function must incentivize three choices

VΩ at (Ω− 1,Ω,Ω + 1)

VΩ+1 at (Ω,Ω + 1,Ω + 2)



Commonly known ω

Ω− 2 Ω− 1 Ω Ω + 1 Ω + 2

1

VΩ+1VΩ

V (y)

Figure: Value functions are approximated byVΩ = (0, 0.29, 0.71, 1, 1) and VΩ+1 = (0, 0, 0.29, 0.71, 1).



Solution p = 1/2, privately known ω

Ω− 2 Ω− 1 Ω Ω + 1 Ω + 2

1

VΩ+1VΩ

V (y)

Figure: Value functions are approximated byVΩ = (0, 0.22, 0.54, 0.80, 0.80) and VΩ+1 = (0, 0, 0.36, 0.79, 1).



Discussion

A solution exists for all values of p ∈ (0, 1).

Some multiplicities for certain values of p in unobserved zcase.

The IC constraint for agent of type Ω + 1 is binding.

Binding IC constraint impacts value functions in an asymmetricway.

Restricts upside when agent chooses VΩ

Increases downside possibility when agent chooses VΩ+1.



Conclusion

Identified an intuitive framework that shows how asymmetries canarise around the reference point for an agent’s value function evenwhen the distribution of choices are symmetric.

Limitations of current examples

Two states doesn’t give full picture of what would happenwith constraints above and below

Finite number of decision states limits discussions aboutconcavity and inflection points

By revealing state that is known to the agent, we are allowingagent the freedom to choose their reference point

This provides a notion of selecting aspiration level


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