Promises and Endogenous Reneging Costs Yuval Heller (Bar Ilan - - PowerPoint PPT Presentation

Promises and Endogenous Reneging Costs

Yuval Heller (Bar Ilan University) David Sturrock (Institute for Fiscal Studies & UCL) Zurich, March 2018

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 1 / 41

Outline

1

Introduction

2

The Partnership Game

3

Evolutionary Analysis

4

Variants and Extensions (Work in Progress)

5

Conclusion

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 2 / 41

Introduction

Outline

1

Introduction Background Motivating Example Brief Overview of Results Related Literature and Contribution

2

The Partnership Game

3

Evolutionary Analysis

4

Variants and Extensions (Work in Progress)

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 3 / 41

Introduction Background

Background

Experimental evidence suggests that people suffer from intrinsic (psychological) costs of lying or of reneging on a promise.

Trust (Ellingsen & Johannesson, 03; Charness & Dufwenberg, 06; Vanberg, 08) Sender-receivers (Gneezy, 05; Hurkens & Kartik, 09; Lundquist et al., 09) Reporting dice’s outcome (Fischbacher & Föllmi-Heusi, 13; Shalvi et al., 11;

Gneezy, Kajackaite & Sobel, 18; Abeler, Nosenzo & Raymond, 18).

Details

The intrinsic costs are increasing in the size of the lie:

Distance between the reported and the true outcome. The damage induced to the partner. How others perceive the agent’s behavior.

In some setups, small lies are normative.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 4 / 41

Introduction Background

Background

Experimental evidence suggests that people suffer from intrinsic (psychological) costs of lying or of reneging on a promise.

Trust (Ellingsen & Johannesson, 03; Charness & Dufwenberg, 06; Vanberg, 08) Sender-receivers (Gneezy, 05; Hurkens & Kartik, 09; Lundquist et al., 09) Reporting dice’s outcome (Fischbacher & Föllmi-Heusi, 13; Shalvi et al., 11;

Gneezy, Kajackaite & Sobel, 18; Abeler, Nosenzo & Raymond, 18).

Details

The intrinsic costs are increasing in the size of the lie:

Distance between the reported and the true outcome. The damage induced to the partner. How others perceive the agent’s behavior.

In some setups, small lies are normative.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 4 / 41

Introduction Motivating Example

Motivating Example

Two busy researchers consider working together on a paper.

1

Stage 1: Each makes a non-enforceable promise regarding future effort.

2

Stage 2: Each chooses his effort level in the joint project.

Effort choices are strategic complements.

Best-reply of each is to exert a bit less effort than their partner. With ‘cheap talk’ no effort will be exerted (bad outcome).

Research Questions:

How does communication (with reneging costs) enable coordination and commitment to high effort? What reneging cost would emerge from an evolutionary process induced by social learning?

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 5 / 41

Introduction Motivating Example

Motivating Example

Two busy researchers consider working together on a paper.

1

Stage 1: Each makes a non-enforceable promise regarding future effort.

2

Stage 2: Each chooses his effort level in the joint project.

Effort choices are strategic complements.

Best-reply of each is to exert a bit less effort than their partner. With ‘cheap talk’ no effort will be exerted (bad outcome).

Research Questions:

How does communication (with reneging costs) enable coordination and commitment to high effort? What reneging cost would emerge from an evolutionary process induced by social learning?

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 5 / 41

Introduction Motivating Example

Motivating Example

Two busy researchers consider working together on a paper.

1

Stage 1: Each makes a non-enforceable promise regarding future effort.

2

Stage 2: Each chooses his effort level in the joint project.

Effort choices are strategic complements.

Best-reply of each is to exert a bit less effort than their partner. With ‘cheap talk’ no effort will be exerted (bad outcome).

Research Questions:

How does communication (with reneging costs) enable coordination and commitment to high effort? What reneging cost would emerge from an evolutionary process induced by social learning?

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 5 / 41

Introduction Brief Overview of Results

Brief Overview of Results

Both low and high reneging costs induce zero efforts. With quadratic payoffs, intermediate level of reneging costs induce:

Maximal promises. Substantial (non-maximal) efforts. Second-best outcome (& better than Stackelberg equilibrium).

Observable costs: The second-best cost is evolutionary stable. Demonstrating robustness to:

Sequential communication. One-sided reneging costs. Discontinuity in the reneging cost around zero. Imperfect observability.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 6 / 41

Introduction Brief Overview of Results

Brief Overview of Results

Both low and high reneging costs induce zero efforts. With quadratic payoffs, intermediate level of reneging costs induce:

Maximal promises. Substantial (non-maximal) efforts. Second-best outcome (& better than Stackelberg equilibrium).

Observable costs: The second-best cost is evolutionary stable. Demonstrating robustness to:

Sequential communication. One-sided reneging costs. Discontinuity in the reneging cost around zero. Imperfect observability.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 6 / 41

Introduction Brief Overview of Results

Brief Overview of Results

Both low and high reneging costs induce zero efforts. With quadratic payoffs, intermediate level of reneging costs induce:

Maximal promises. Substantial (non-maximal) efforts. Second-best outcome (& better than Stackelberg equilibrium).

Observable costs: The second-best cost is evolutionary stable. Demonstrating robustness to:

Sequential communication. One-sided reneging costs. Discontinuity in the reneging cost around zero. Imperfect observability.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 6 / 41

Introduction Brief Overview of Results

Brief Overview of Results

Both low and high reneging costs induce zero efforts. With quadratic payoffs, intermediate level of reneging costs induce:

Maximal promises. Substantial (non-maximal) efforts. Second-best outcome (& better than Stackelberg equilibrium).

Observable costs: The second-best cost is evolutionary stable. Demonstrating robustness to:

Sequential communication. One-sided reneging costs. Discontinuity in the reneging cost around zero. Imperfect observability.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 6 / 41

Introduction Related Literature and Contribution

Related Literature and Contributions

Communication with lying costs

(e.g., Kartik, Ottaviani & Squintani, 2007; Kartik, 2009). Our contributions: (i) bilateral communication, (ii) communication about future action, (III) endogenising intrinsic costs.

Partnership Games (e.g., Holmstrom, 82; Radner, Myerson & Maskin, 86)

We demonstrate a novel mechanism to sustain efficient outcomes.

Commitment in strategic encounters (e.g., Schelling, 1960; Caruana &

Einav, 2008; Ellingsen & Miettinen, 2008). In most existing models more commitment is advantageous. In our model, the optimal level of commitment is intermediate.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 7 / 41

Introduction Related Literature and Contribution

Related Literature and Contributions

Communication with lying costs

(e.g., Kartik, Ottaviani & Squintani, 2007; Kartik, 2009). Our contributions: (i) bilateral communication, (ii) communication about future action, (III) endogenising intrinsic costs.

Partnership Games (e.g., Holmstrom, 82; Radner, Myerson & Maskin, 86)

We demonstrate a novel mechanism to sustain efficient outcomes.

Commitment in strategic encounters (e.g., Schelling, 1960; Caruana &

Einav, 2008; Ellingsen & Miettinen, 2008). In most existing models more commitment is advantageous. In our model, the optimal level of commitment is intermediate.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 7 / 41

Introduction Related Literature and Contribution

Related Literature and Contributions

Communication with lying costs

(e.g., Kartik, Ottaviani & Squintani, 2007; Kartik, 2009). Our contributions: (i) bilateral communication, (ii) communication about future action, (III) endogenising intrinsic costs.

Partnership Games (e.g., Holmstrom, 82; Radner, Myerson & Maskin, 86)

We demonstrate a novel mechanism to sustain efficient outcomes.

Commitment in strategic encounters (e.g., Schelling, 1960; Caruana &

Einav, 2008; Ellingsen & Miettinen, 2008). In most existing models more commitment is advantageous. In our model, the optimal level of commitment is intermediate.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 7 / 41

Introduction Related Literature and Contribution

Related Literature and Contributions

Indirect evolutionary approach (e.g., Guth & Yaari,1992; Ok &

Vega-Redondo , 2001; Dekel, Ely & Yilnkaya, 2007; Heifetz et al., 2007; Alger & Weibull, 2013). We apply this approach to study the evolution of preferences for breaking commitments.

Experimental evidence of truth-telling and promise-keeping (e.g.,

Fischbacher & Föllmi-Heusi, 2013; Gneezy, Kajackaite & Sobel, 2018; Abeler, Nosenzo & Raymond, forthcoming). We present an evolutionary foundation for intermediate lying/reneging costs.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 8 / 41

Introduction Related Literature and Contribution

Related Literature and Contributions

Indirect evolutionary approach (e.g., Guth & Yaari,1992; Ok &

Vega-Redondo , 2001; Dekel, Ely & Yilnkaya, 2007; Heifetz et al., 2007; Alger & Weibull, 2013). We apply this approach to study the evolution of preferences for breaking commitments.

Experimental evidence of truth-telling and promise-keeping (e.g.,

Fischbacher & Föllmi-Heusi, 2013; Gneezy, Kajackaite & Sobel, 2018; Abeler, Nosenzo & Raymond, forthcoming). We present an evolutionary foundation for intermediate lying/reneging costs.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 8 / 41

The Partnership Game

Outline

1

Introduction

2

The Partnership Game Supermodular Payoffs No-Effort Result Quadratic Payoffs Analysis

3

Evolutionary Analysis

4

Variants and Extensions (Work in Progress)

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 9 / 41

The Partnership Game Supermodular Payoffs

The Partnership Game

Two players, two stages. Each player i is endowed with reneging cost of λi ≥ 0. Stage 1: each player i sends message si ∈ [0,1]. Stage 2: each player chooses his effort level xi ∈ [0,1] Choices at each stage are simultaneous.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 10 / 41

The Partnership Game Supermodular Payoffs

The Partnership Game

Two players, two stages. Each player i is endowed with reneging cost of λi ≥ 0. Stage 1: each player i sends message si ∈ [0,1]. Stage 2: each player chooses his effort level xi ∈ [0,1] Choices at each stage are simultaneous.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 10 / 41

The Partnership Game Supermodular Payoffs

Material Payoff

Material payoff π(xi,xj):

continuously differentiable. weakly supermodular:

∂ 2π(xi,xj) ∂xi∂xj

≥ 0. encourages shirking: ∂π(xi,xj )

∂xi

• xj =xi>0 < 0.

Subjective payoff includes reneging costs: U(xi,xj,si,λi) = π(xi,xj)−λi ·D (|si −xi|)

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 11 / 41

The Partnership Game Supermodular Payoffs

Material Payoff

Material payoff π(xi,xj):

continuously differentiable. weakly supermodular:

∂ 2π(xi,xj) ∂xi∂xj

≥ 0. encourages shirking: ∂π(xi,xj )

∂xi

• xj =xi>0 < 0.

Subjective payoff includes reneging costs: U(xi,xj,si,λi) = π(xi,xj)−λi ·D (|si −xi|)

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 11 / 41

The Partnership Game Supermodular Payoffs

Intrinsic Reneging Cost

Assumption: The cost function D (|si −xi|) is strictly increasing in the size of the lie |si −xi|. |si −xi| can also represent social identity (i.e., the likelihood that

• thers perceive the agent as liar), in a setup in which
• bserved effort = intended effort + noise.

We allow discontinuities in the reneging costs (e.g., at zero). We assume that exerting more effort than promised is costly.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 12 / 41

The Partnership Game Supermodular Payoffs

Intrinsic Reneging Cost

Assumption: The cost function D (|si −xi|) is strictly increasing in the size of the lie |si −xi|. |si −xi| can also represent social identity (i.e., the likelihood that

• thers perceive the agent as liar), in a setup in which
• bserved effort = intended effort + noise.

We allow discontinuities in the reneging costs (e.g., at zero). We assume that exerting more effort than promised is costly.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 12 / 41

The Partnership Game Supermodular Payoffs

Intrinsic Reneging Cost

Assumption: The cost function D (|si −xi|) is strictly increasing in the size of the lie |si −xi|. |si −xi| can also represent social identity (i.e., the likelihood that

• thers perceive the agent as liar), in a setup in which
• bserved effort = intended effort + noise.

We allow discontinuities in the reneging costs (e.g., at zero). We assume that exerting more effort than promised is costly.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 12 / 41

The Partnership Game No-Effort Result

No Efforts if Reneging Costs are too High / too Low

Proposition

For each ε > 0, there exist λ ε > λ ε > 0, such that in any equilibrium both agents exert at most ε effort if either (1) λi,λj < λ ε, or (2) λi,λj > λ ε.

Sketch of Proof.

Low λs: weak commitment power cannot stop 2nd-round shirking. High λs: too strong commitment power ⇒ agents shirk in their promises in the 1st round.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 13 / 41

The Partnership Game No-Effort Result

No Efforts if Reneging Costs are too High / too Low

Proposition

For each ε > 0, there exist λ ε > λ ε > 0, such that in any equilibrium both agents exert at most ε effort if either (1) λi,λj < λ ε, or (2) λi,λj > λ ε.

Sketch of Proof.

Low λs: weak commitment power cannot stop 2nd-round shirking. High λs: too strong commitment power ⇒ agents shirk in their promises in the 1st round.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 13 / 41

The Partnership Game No-Effort Result

No Efforts if Reneging Costs are too High / too Low

The no-effort result depends on simultaneous communication + “two-sided” reneging costs. In the remaining analysis we focus, for tractability, on quadratic payoffs. (Partial) robustness of result: With sequential communication and/or “one-sided” reneging costs, high λs don’t induce zero efforts; yet, they induce less efforts than those induced by intermediate λs.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 14 / 41

The Partnership Game Quadratic Payoffs

Quadratic Payoffs

Material payoff π(xi,xj): π (xi,xj) = xi ·xj − c ·x2

i

2 . xi ·xj: Both players obtain the same gross return from the project.

c·x2

i

2 : Each agent incurs a convex cost of their own effort.

Parameter c > 1 governing the cost of effort.

Socially optimal outcome: maximal efforts (c < 2).

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 15 / 41

The Partnership Game Quadratic Payoffs

Quadratic Payoffs

Subjective payoffs also include intrinsic reneging costs: Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·(si −xi)2

Reneging cost is the square of the difference between the promise and the actual effort. Strict concavity: all subgame-perfect equilibria are pure.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 16 / 41

The Partnership Game Quadratic Payoffs

Quadratic Payoffs

Subjective payoffs also include intrinsic reneging costs: Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·(si −xi)2

Reneging cost is the square of the difference between the promise and the actual effort. Strict concavity: all subgame-perfect equilibria are pure.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 16 / 41

The Partnership Game Analysis

Solving the 2nd Round

Best reply in the 2nd round: x∗

i (xj,si,λi,c) = xj +λi ·si

c +λi Best reply is a weighted average of xj

c and si.

If λi = 0: Best-reply is x∗

i = xj c < xj.

If λi = ∞: Best-reply is x∗

i = si.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 17 / 41

The Partnership Game Analysis

Solving the 2nd Round

Simultaneously solving the best-response functions yields a unique 2nd-stage equilibrium: xe

i (si,sj,λi,λj,c) =

λi ·(c +λj)·si +λj·sj λi ·(c +λj)+λj +(c −1)·(λj +c +1) Effort xe

i undercuts a weighted average of si and sj.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 18 / 41

The Partnership Game Analysis

1st Round Best Replies

Substituting the unique 2nd-stage equilibrium strategies yields utility Ui(si,sj,c) as a function of the 1st-stage messages. Best-reply is linear in the partner’s message. Three regions for the best-reply function (see next slide).

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 19 / 41

The Partnership Game Analysis

1st Round Best Replies

(Example c=1.2)

Partner’s cost is high (red): agent undercuts the partner’s message. Partner’s cost is intermediate (yellow): agent overcuts the partner. Partner’s reneging cost is low (blue): agent sends maximal message.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 20 / 41

The Partnership Game Analysis

Unique Subgame Perfect Equilibrium

Theorem

A unique subgame perfect equilibrium. There are 3 regions:

1 Both costs are either (1) too high, or (2) too low (red): minimal

messages and minimal efforts (si = sj = xi = xj = 0).

2 A convex symmetric region with intermediate costs (blue): maximal

messages (si = sj = 1), substantial non-maximal efforts.

3 One agent has high cost and the partner has a low cost (yellow):

“Stackelberg-like” equilibrium – only the agent with the higher cost makes a maximal promise (si = 1 > sj). We ignore a trembling-hand imperfect equilibrium with si = sj = 0.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 21 / 41

The Partnership Game Analysis

Unique Subgame Perfect Equilibrium

1 Both costs are (1) too high, or (2) too low (red): minimal messages. 2 Convex region of intermediate costs (blue): maximal messages. 3 One cost high, other cost low (yellow): one message is maximal.

(Example c=1.2)

• ther c-s

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 22 / 41

The Partnership Game Analysis

The Highest Intermediate Cost

• λ +

C ,λ + C

• The intermediate (blue) region is non-empty for each c ∈ (1,1.25).

The highest point in this region

• λ +

C ,λ + C

• induces the “second-best”
• utcome: the highest sum of material payoffs.

(i.e., π

• λ +

ρ ,λ + ρ

• > π
• λ

′,λ ′

∀λ ′ = λ +

ρ .).

The sum of payoffs is higher than the Stackelberg equilibrium of sequential efforts. The sum of payoffs is increasing in c, and converges to the first-best

• utcome (maximal efforts) as c → 1.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 23 / 41

Evolutionary Analysis

Outline

1

Introduction

2

The Partnership Game

3

Evolutionary Analysis

4

Variants and Extensions (Work in Progress)

5

Conclusion

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 24 / 41

Evolutionary Analysis

Evolution of Observable Reneging Costs

We follow the ‘indirect’ evolutionary approach whereby players make choices to maximise their utility, but payoffs from the ‘underlying’ game determine fitness (see, e.g., Guth and Yaari, 1992). Continuum of agents. Each agent is endowed with a reneging cost (type). Agents are randomly matched into pairs. Each player observes their partner’s reneging cost before playing. Each pair plays the unique perfect equilibrium of the partnership game, given their reneging costs.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 25 / 41

Evolutionary Analysis

Evolution of Observable Reneging Costs

We follow the ‘indirect’ evolutionary approach whereby players make choices to maximise their utility, but payoffs from the ‘underlying’ game determine fitness (see, e.g., Guth and Yaari, 1992). Continuum of agents. Each agent is endowed with a reneging cost (type). Agents are randomly matched into pairs. Each player observes their partner’s reneging cost before playing. Each pair plays the unique perfect equilibrium of the partnership game, given their reneging costs.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 25 / 41

Evolutionary Analysis

The Population Game

Agent’s fitness, π, is their utility, excluding reneging costs: πi(xi,xj,c) = xi ·xj − c·x2

i

2 .

That is, πc (λi,λj) = xe

i ·xe j − c·(xe

i ) 2

2

. πc defines a symmetric population game Γc = (R+,πc). A social learning process in which more successful types (reneging costs) become more frequent.

Fitness-monotone dynamics (e.g., as the replicator dynamics, Taylor & Jonker, 1978).

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 26 / 41

Evolutionary Analysis

The Population Game

Agent’s fitness, π, is their utility, excluding reneging costs: πi(xi,xj,c) = xi ·xj − c·x2

i

2 .

That is, πc (λi,λj) = xe

i ·xe j − c·(xe

i ) 2

2

. πc defines a symmetric population game Γc = (R+,πc). A social learning process in which more successful types (reneging costs) become more frequent.

Fitness-monotone dynamics (e.g., as the replicator dynamics, Taylor & Jonker, 1978).

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 26 / 41

Evolutionary Analysis

Stable Population States

It is well known that stable population states correspond to Nash equilibria of the population game (e.g., Weibull, 1995; Cressman, 1997;

Sandholm, 2010):

1 Any symmetric strict equilibrium corresponds to a stable population

state.

2 Any stable population state corresponds to a symmetric Nash

equilibrium.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 27 / 41

Evolutionary Analysis

Stability of Intermediate Reneging Costs

Theorem

Assume c ∈ (1,1.24). The highest cost λ +

c for which both players send

maximal promises is the unique pure symmetric Nash/strict equilibrium.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 28 / 41

Variants and Extensions (Work in Progress)

Outline

1

Introduction

2

The Partnership Game

3

Evolutionary Analysis

4

Variants and Extensions (Work in Progress) Sequential Communication One-Sided Reneging Costs Discontinuity in Reneging Costs Around Zero Evolution under Partial Observability

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 29 / 41

Variants and Extensions (Work in Progress) Sequential Communication

Sequential Communication

Promises are communicated sequentially. One of the agents is randomly chosen to be the first. Unique subgame perfect equilibrium for each pair of λs. The unique equilibrium in the baseline case remains the same whenever at least one of the players sends a maximal message (blue and yellow regions). In the red region (two high λs): unique SPE in which the first player send maximal message, and the partner undercuts him. λ +

c is remains evolutionary stable, and it still induces the second-best

• utcome.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 30 / 41

Variants and Extensions (Work in Progress) Sequential Communication

Sequential Communication

Promises are communicated sequentially. One of the agents is randomly chosen to be the first. Unique subgame perfect equilibrium for each pair of λs. The unique equilibrium in the baseline case remains the same whenever at least one of the players sends a maximal message (blue and yellow regions). In the red region (two high λs): unique SPE in which the first player send maximal message, and the partner undercuts him. λ +

c is remains evolutionary stable, and it still induces the second-best

• utcome.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 30 / 41

Variants and Extensions (Work in Progress) Sequential Communication

Sequential Communication

The unique equilibrium is unchanged in the blue and yellow regions. In the red region (two high λs): unique SPE in which the first player send maximal message, and the partner undercuts him. λ +

c is (1) evolutionary stable, and (2) induces the 2nd-best outcome.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 31 / 41

Variants and Extensions (Work in Progress) One-Sided Reneging Costs

One-Sided Reneging Costs

Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·1(si>xi) ·(si −xi)2

Agent suffers a reneging cost only when effort < promise. May reflect cost that is proportional to the damage induced to the partner by the lie.

There is no damage when effort > promise. Damage is xj ·(si −xi) when effort < promise.

Unique equilibrium remains the same in the blue and yellow regions. Asymmetric pure equilibria in the red region (two high λs): One agent promising one, the partner promises a low effort (+ inefficient symmetric mixed equilibrium). λ +

c remains evolutionary stable, and it induces the 2nd-best outcome.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 32 / 41

Variants and Extensions (Work in Progress) One-Sided Reneging Costs

One-Sided Reneging Costs

Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·1(si>xi) ·(si −xi)2

Agent suffers a reneging cost only when effort < promise. May reflect cost that is proportional to the damage induced to the partner by the lie.

There is no damage when effort > promise. Damage is xj ·(si −xi) when effort < promise.

Unique equilibrium remains the same in the blue and yellow regions. Asymmetric pure equilibria in the red region (two high λs): One agent promising one, the partner promises a low effort (+ inefficient symmetric mixed equilibrium). λ +

c remains evolutionary stable, and it induces the 2nd-best outcome.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 32 / 41

Variants and Extensions (Work in Progress) One-Sided Reneging Costs

One-Sided Reneging Costs

The equilibrium remains the same in the blue and yellow regions. Asymmetric pure equilibria in the red region (two high λs): One agent promising one, the partner promises a low effort. λ +

c remains evolutionary stable, and it induces the 2nd-best outcome.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 33 / 41

Variants and Extensions (Work in Progress) Discontinuity in Reneging Costs Around Zero

Discontinuity in Reneging Costs Around Zero

Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·(si −xi)2 −γi ·1Si=xi.

Fixed cost of γi for any lie (regardless of its size). Fixed small γi < ¯ γc:

λ +

c remains evolutionary stable, and it induces the same 2nd-best

• utcome.

”Sunk” cost of γi is subtracted from the subjective payoff of agent i.

Fixed large γi > ¯ γc:

Agents exert the same effort as promised. 1st round: Each agent undercuts the partner’s promise. Unique equilibrium: no efforts.

Endogenous γi : The set of distributions over pairs

• (λ +

c ,γ)|γ ≤ ¯

γc

• is

evolutionary stable.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 34 / 41

Variants and Extensions (Work in Progress) Discontinuity in Reneging Costs Around Zero

Discontinuity in Reneging Costs Around Zero

Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·(si −xi)2 −γi ·1Si=xi.

Fixed cost of γi for any lie (regardless of its size). Fixed small γi < ¯ γc:

λ +

c remains evolutionary stable, and it induces the same 2nd-best

• utcome.

”Sunk” cost of γi is subtracted from the subjective payoff of agent i.

Fixed large γi > ¯ γc:

Agents exert the same effort as promised. 1st round: Each agent undercuts the partner’s promise. Unique equilibrium: no efforts.

Endogenous γi : The set of distributions over pairs

• (λ +

c ,γ)|γ ≤ ¯

γc

• is

evolutionary stable.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 34 / 41

Variants and Extensions (Work in Progress) Discontinuity in Reneging Costs Around Zero

Discontinuity in Reneging Costs Around Zero

Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·(si −xi)2 −γi ·1Si=xi.

Fixed cost of γi for any lie (regardless of its size). Fixed small γi < ¯ γc:

λ +

c remains evolutionary stable, and it induces the same 2nd-best

• utcome.

”Sunk” cost of γi is subtracted from the subjective payoff of agent i.

Fixed large γi > ¯ γc:

Agents exert the same effort as promised. 1st round: Each agent undercuts the partner’s promise. Unique equilibrium: no efforts.

Endogenous γi : The set of distributions over pairs

• (λ +

c ,γ)|γ ≤ ¯

γc

• is

evolutionary stable.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 34 / 41

Variants and Extensions (Work in Progress) Discontinuity in Reneging Costs Around Zero

Discontinuity in Reneging Costs Around Zero

Ui(xi,xj,si,c) = xi ·xj − c·x2

i

2 − λi 2 ·(si −xi)2 −γi ·1Si=xi.

Fixed cost of γi for any lie (regardless of its size). Fixed small γi < ¯ γc:

λ +

c remains evolutionary stable, and it induces the same 2nd-best

• utcome.

”Sunk” cost of γi is subtracted from the subjective payoff of agent i.

Fixed large γi > ¯ γc:

Agents exert the same effort as promised. 1st round: Each agent undercuts the partner’s promise. Unique equilibrium: no efforts.

Endogenous γi : The set of distributions over pairs

• (λ +

c ,γ)|γ ≤ ¯

γc

• is

evolutionary stable.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 34 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Evolution with Partial Observability

The main result relies on a strong assumption, namely, that each agent perfectly observes their partner’s reneging cost. Can we extend the main result to a setup in which the partner’s reneging cost is observed with a high probability? Let q ∈ [0,1] denote the fraction of matches in which each player

• bserves their partner’s reneging cost.

Simplifying assumption of perfect correlation within each pair: either both agents observe each other’s type, or neither of them does.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 35 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Evolution with Partial Observability

The main result relies on a strong assumption, namely, that each agent perfectly observes their partner’s reneging cost. Can we extend the main result to a setup in which the partner’s reneging cost is observed with a high probability? Let q ∈ [0,1] denote the fraction of matches in which each player

• bserves their partner’s reneging cost.

Simplifying assumption of perfect correlation within each pair: either both agents observe each other’s type, or neither of them does.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 35 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Evolution with Partial Observability

Consider a setup in which all incumbents have cost λ. A ‘mutant’ is endowed with λ ′. With probability 1−q, the mutant’s partner (wrongly) believes that he faces an incumbent, and plays his part of the equilibrium of G (λ,λ). Mutant best replies to it. With probability q, each agent observes the partner’s cost, and they play the equilibrium of G (λ,λ ′).

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 36 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Evolution with Partial Observability

Consider a setup in which all incumbents have cost λ. A ‘mutant’ is endowed with λ ′. With probability 1−q, the mutant’s partner (wrongly) believes that he faces an incumbent, and plays his part of the equilibrium of G (λ,λ). Mutant best replies to it. With probability q, each agent observes the partner’s cost, and they play the equilibrium of G (λ,λ ′).

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 36 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Robustness of Main Result

Definition

Population state λ is a pure (strict) equilibrium of the population game, if any mutant is weakly (strictly) outperformed by the incumbents.

Theorem (Robustness of Main Result)

λ +

c is a symmetric strict equilibrium for any q < 1 that is sufficiently close

to one. Intuition: first-order loss of a deviator with a slightly smaller reneging cost is strictly positive.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 37 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Robustness of Main Result

Definition

Population state λ is a pure (strict) equilibrium of the population game, if any mutant is weakly (strictly) outperformed by the incumbents.

Theorem (Robustness of Main Result)

λ +

c is a symmetric strict equilibrium for any q < 1 that is sufficiently close

to one. Intuition: first-order loss of a deviator with a slightly smaller reneging cost is strictly positive.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 37 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Non-Robustness of No Effort Equilibrium

Theorem (No Observability ⇒ No Effort)

If q = 0, then there are no efforts in any Nash equilibrium. In line with a main stylized result in the literature on evolution of preferences (see, e.g., Ok & Vega-Redondo, 2001). Intuition: if there were positive efforts, a mutant with zero lying costs would strictly outperform the incumbents. Is the above result robust to low levels of observability?

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 38 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Non-Robustness of No Effort Equilibrium

Theorem (No Observability ⇒ No Effort)

If q = 0, then there are no efforts in any Nash equilibrium. In line with a main stylized result in the literature on evolution of preferences (see, e.g., Ok & Vega-Redondo, 2001). Intuition: if there were positive efforts, a mutant with zero lying costs would strictly outperform the incumbents. Is the above result robust to low levels of observability?

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 38 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Non-Robustness of No Effort Equilibrium

Any amount of observability induces positive effort

Theorem

Fix any q > 0. Then, in any symmetric pure Nash equilibrium, players exert positive levels of effort on the equilibrium path. Intuition: if a population is playing a ‘no effort’ equilibrium, there is always a mutant who exerts effort against the incumbents and does better.

Argument depends on the observability being sufficiently correlated.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 39 / 41

Variants and Extensions (Work in Progress) Evolution under Partial Observability

Non-Robustness of No Effort Equilibrium

Any amount of observability induces positive effort

Theorem

Fix any q > 0. Then, in any symmetric pure Nash equilibrium, players exert positive levels of effort on the equilibrium path. Intuition: if a population is playing a ‘no effort’ equilibrium, there is always a mutant who exerts effort against the incumbents and does better.

Argument depends on the observability being sufficiently correlated.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 39 / 41

Conclusion

Outline

1

Introduction

2

The Partnership Game

3

Evolutionary Analysis

4

Variants and Extensions (Work in Progress)

5

Conclusion

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 40 / 41

Conclusion

Conclusion

Partnership games: supermodular payoffs, shirking is a best reply. Agents make non-binding promises (with intrinsic reneging costs). Too high and too low reneging costs do not induce efforts. Intermediate level of reneging costs (with quadratic payoffs):

induces the 2nd-best outcome: maximal promises, substantial efforts. evolutionary stable (when costs are observable).

Main results are robust to sequential communication, one-sided reneging costs, discontinuity of costs around zero, imperfect

• bservability of costs.

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 41 / 41

Backup Slides

Backup Slides

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 42 / 41

Backup Slides

Experimental Results on Dice Rolling

Subjects report outcome of a dice roll, and get money proportional to the reported amount. Dice outcome can never be observed: 40% of subjects lie. Less than half of liars report 6. Dice outcome can be observed ex-post by experimenter (while keeping anonymity):

Only 20% of subjects lie; most liars got low outcomes, and report 6.

Fischbacher & Föllmi-Heusi (13); Shalvi et al. (11); Gneezy, Kajackaite & Sobel (18); Abeler, Nosenzo & Raymond (18).

Back Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 43 / 41

Backup Slides

Equilibria Regions for Various Values of c

Back

c=1.24 c=1.15

Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 44 / 41

Backup Slides

The Partnership Game

Substituting the unique 2nd-stage equilibrium strategies yields utility Ui(si,sj,c) as a function of the 1st-stage message: Ui(si,sj,c) =[(c +λj)λisi +λjsj][(c +λi)λjsj +λisi] [(c +λi)(c +λj)−1]2 − c[(c +λj)λisi +λjsj]2 2[(c +λi)(c +λj)−1]2 − λi 2

• si − (c +λj)λisi +λjsj

(c +λi)(c +λj)−1 2

Back Heller and Sturrock Promises & Endogenous Reneging Costs Zurich, March 2018 41 / 41