Trust and adaptive learning in implicit contracts

Abstract

Trust is a phenomenon that still is quite rarely investigated in agency theory. According to a common intuitive reasoning, trust should develop over time and it should evolve even in finite implicit-contract relationships. However, if the contracting parties are fully rational, theory cannot explain this. We therefore extend the standard model and develop a model of a finite relationship where the principal promises to pay a voluntary period-by-period bonus if the agent has worked according to the implicit agreement. The agent is boundedly rational and unable to foresee the principal’s future bonus decisions. The principal is, with some probability, honest and pays a promised bonus even in situations where ex-post cheating would be optimal. Based on the agent’s adaptive learning process, we show how trust evolves depending on the principal’s bonus-payment strategy. Depending on different levels of the agent’s bounded rationality, we derive the principal’s optimal pure strategy as part of a unique equilibrium. In an extension we show that the results are robust if the agent has bounded recall. The optimal strategy pattern mirrors a subset of trigger strategies which is exogenous in the standard model. Our findings imply that subjective incentives are more effective with increasing tenure of employees, or, that the optimal level of trust depends on how fast work environments change.

Appendix

Proof of Proposition 1

• (a) Differentiating ${\mathcal{S}}\left( \varvec{ \theta }\right)$ as given by Eq. 13 with respect to δ we obtain $\frac{\partial {\mathcal{S}}\left( \varvec{ \theta }\right) }{\partial \delta } =\sum_{t=1}^{T}S_{t}\left( \varvec{ \theta }\right) \left( t-1\right) \delta ^{t-2}>0$. As $\frac{\partial {\delta }} {\partial r}=-i\left( 1+r\right) ^{-\left( i+1\right) }<0$ it follows $\frac{\partial {\mathcal{S}}\left( \varvec{ \theta }\right) }{\partial r}<0$. Let $\varvec{ \theta }^{\ast }$ be the optimal strategy with r and $\varvec{ \theta }^{\prime }$ the optimal strategy with r′ > r. As ${\mathcal{S}}\left( \varvec{ \theta }\right)$ is decreasing in $r,{\mathcal{S}}\left( \varvec{ \theta }^{\prime },r^{\prime }\right) <{\mathcal{S}}\left( \varvec{ \theta }^{\prime },r\right),$ and as ${\mathcal{S}}\left( \varvec{ \theta }^{\prime },r\right) <{\mathcal{S }}\left( \varvec{ \theta }^{\ast },r\right)$ it follows ${\mathcal{S}} \left( \varvec{ \theta }^{\prime },r^{\prime }\right) <{\mathcal{S}}\left( \varvec{ \theta }^{\ast },r\right)$ and the principal’s equilibrium payoff decreases in r.

• (b) Consider period t with induced trust γ _t−1. If non-payment is optimal in period t with interest rate r, given γ _t−1, then non-payment must be optimal in period t with r′ > r given γ _t−1, too, as the surplus function (or equivalently: the left hand side of the non-reneging constraint Eq. 12) is decreasing in r. Hence, starting with the same initial level of trust, γ₀, the induced level of trust in period t with interest rate r, γ ^r _t−1 , must be at least as high as with r′ > r, i.e., γ ^r _t−1  ≥ $\gamma _{{t - 1}}^{{r^{\prime } }}$ for all t; for γ ^r _t−1  < $\gamma _{{t - 1}}^{{r^{\prime } }}$  to be true for some period t there must have been a period τ < t with γ ^r _τ−1  = $\gamma _{{t - 1}}^{{r^{\prime } }}$ and with non-payment for r but payment for r′: a contradiction. Hence, the number of periods $\sum_{i=1}^{t}\theta _{i}$ in which the bonus is paid up to some period t is weakly decreasing in r.

• (c) A higher level of ex ante trust increases every summand in Eq. 13 such that ${\mathcal{S}}$ is increasing in γ₀. $\square$

Proof of Lemma 1

Writing payoffs S _t as a function of induced trust γ _t−1 at the beginning of period t transforms Eq. 13 into ${\mathcal{ S}}^{D}=\sum_{t=1}^{T}S_{t}^{D}(\gamma _{t-1}^{{\bf 0}})\delta ^{t-1}$ for strict non-payment, $\varvec{\theta}={\bf 0}$, and ${\mathcal{S}} ^{H}=\sum_{t=1}^{T}S_{t}^{H}(\gamma _{t-1}^{{\bf 1}})\delta ^{t-1}$ for strict payment, $\varvec{\theta}={\bf 1},$ where $\gamma _{t-1}^{\varvec{ \theta } }$ indicates the level of trust at the beginning of period t contingent on $\varvec{\theta}\in \left\{ {\bf 0,1}\right\}$. To show how ${\mathcal{S}}^{D}$ and ${\mathcal{S}}^{H}$ behave depending on the principal’s lifetime T we write them explicitly as a function of T, ${\mathcal{S}} ^{D}(T)$ and ${\mathcal{S}}^{H}(T)$. For expositional brevity assume T is continuous. Surpluses become ${\mathcal{S}}^{D}(T)=\int_{t=1}^{T}S_{t}^{D}( \gamma _{t-1}^{{\bf 0}})\delta ^{t-1} dt$ and ${\mathcal{S}} ^{H}(T)=\int_{t=1}^{T}S_{t}^{H}(\gamma _{t-1}^{{\bf 1}})\delta ^{t-1}dt$. The first derivatives with respect to T obtain as

$$ {\frac{d{\mathcal{S}}^{D}(T)}{dT}}=S_{T}^{D}(\gamma _{T-1}^{{\bf 0}})\delta ^{T-1}>0;\quad {\frac{d{\mathcal{S}}^{H}(T)}{dT}}=S_{T}^{H}(\gamma _{T-1}^{{\bf 1}})\delta ^{T-1}>0; $$

and second derivatives as

$$ \begin{aligned} {\frac{d^{2}{\mathcal{S}}^{D}(T)}{dT^{2}}}\,=\,&\delta ^{T-1}{\frac{dS_{T}^{D}}{ d\gamma _{T-1}^{{\bf 0}}}}\frac{d\gamma _{T-1}^{{\bf 0}}}{dT} +S_{T}^{D}(\gamma _{T-1}^{{\bf 0}})\delta ^{T-1}\ln \left( \delta \right) \\ {\frac{d^{2}{\mathcal{S}}^{H}(T)}{dT^{2}}}\,=\,&\delta ^{T-1}{\frac{dS_{T}^{H}}{ d\gamma _{T-1}^{{\bf 1}}}}\frac{d\gamma _{T-1}^{{\bf 1}}}{dT} +S_{T}^{H}(\gamma _{T-1}^{{\bf 1}})\delta ^{T-1}\ln \left( \delta \right). \end{aligned} $$

Notice that according to (1)

$$ \begin{aligned} \gamma _{T-1}^{{\bf 0}}&={\frac{\alpha }{\alpha +\beta +T-1}} \\ \gamma _{T-1}^{{\bf 1}}&={\frac{\alpha +T-1}{\alpha +\beta +T-1}} \end{aligned} $$

such that $\frac{d\gamma _{T-1}^{{\bf 0}}}{dT}=-{\frac{\alpha }{\left( \alpha +\beta +T-1\right) ^{2}}}<0$ and $\frac{d\gamma _{T-1}^{{\bf 1}}}{dT}={\frac{\beta }{\left( \alpha +\beta +T-1\right) ^{2}}}>0$. Furthermore, we know from Eqs. 14 and 15 that ${\frac{dS_{T}^{D}}{ d\gamma _{T-1}^{{\bf 0}}}}>0$ and ${\frac{dS_{T}^{H}}{d\gamma _{T-1}^{ {\bf 1}}}}>0$. Assuming r = 0, i.e., δ = 1 leading to ln(δ) = 0, it follows that ${\mathcal{S}}^{D}(T)$ is a strictly monotone increasing concave function of T and ${\mathcal{S}}^{H}(T)$ is a strictly monotone increasing convex function of T. Furthermore we know that ${\mathcal{S}}^{D}(1)>{\mathcal{S}}^{H}(1)$ and $\lim_{T=\infty }{\frac{d {\mathcal{S}}^{D}(T)}{dT}}=0$ and $\lim_{T=\infty }{\frac{d{\mathcal{S}}^{H}(T)}{dT }}=1/2$. Hence, there exists a threshold value for T such that strict payment dominates strict non-payment. The same result applies for positive but sufficiently small r.$\square$

Proof of Proposition 2

Assume r = 0 and consider strategy $\varvec{ \theta }^{\prime }=\left(\ldots,0,1,\ldots \right)$ where the principal does not pay the bonus in period τ but does pay it in τ + 1. Now consider strategy $\varvec{ \theta } =\left( \ldots,1,0,\ldots\right)$ where the principal pays in τ and does not pay in period τ + 1. Everything else equal we can compare strategies $\varvec{ \theta }^{\prime }$ and $\varvec{ \theta }$ by comparing their ex ante expected payoffs from period τ and τ + 1. We obtain

$$ \begin{aligned} {\mathcal{S}}(\varvec{ \theta })&=S_{\tau }^{H}\left( \gamma _{\tau -1}\right) +S_{\tau +1}^{D}\left( \gamma _{\tau }\right) \\ {\mathcal{S}}(\varvec{ \theta }^{\prime })&=S_{\tau }^{D}\left( \gamma _{\tau -1}\right) +S_{\tau +1}^{H}\left( \gamma _{\tau }^{\prime }\right). \end{aligned} $$

γ_τ−1  is the level of trust induced at the beginning of period τ, γ_τ(γ ^′_τ ) is the level of trust at the beginning of period τ + 1 given payment (non-payment) in τ. We know that γ_τ > γ_τ −1 > γ ^′_τ . The difference of expected payoffs is

$$ {\mathcal{S}}(\varvec{ \theta })-{\mathcal{S}}(\varvec{ \theta }^{\prime })=\left[ S_{\tau }^{H}\left( \gamma _{\tau -1}\right) -S_{\tau +1}^{H}\left( \gamma _{\tau }^{\prime }\right) \right] +\left[ S_{\tau +1}^{D}\left( \gamma _{\tau }\right) -S_{\tau }^{D}\left( \gamma _{\tau -1}\right) \right] $$

As S ^D and S ^H are increasing in γ, both brackets [·] are strictly positive and therefore ${\mathcal{S}}(\varvec{ \theta })-{\mathcal{S}}(\varvec{ \theta }^{\prime })>0$. By induction, this argument also holds for any number τ of non-payments in a row. As long as r is sufficiently low the same logic applies for r > 0. Hence, if discounting is low and the principal’s optimal strategy exhibits τ payments, the optimal strategy pattern is $\varvec{ \theta }^{\ast }=(\theta _{1}=1,\theta _{2}=1,\ldots,\theta _{\tau }=1,\theta _{\tau +1}=0,\ldots,\theta _{T}=0)$: Trust will be raised to its desired maximum until period τ and in the remaining periods the gains from trust formation will be earned by non-payments.$\square$

Proof of Proposition 3

• (a) The surplus Eq. 13 for strategy $\varvec{ \theta }^{\ast }=(\theta _{1}=1,\theta _{2}=1,\ldots,\theta _{T-1}=1,\theta _{T}=0)$ obtains as

  $$ \begin{aligned} {\mathcal{S}}(\varvec{ \theta }^{\ast }) =\,&S_{1}^{H}(\gamma _{0})+\sum_{t=2}^{T-1}S_{i}^{H}(\gamma _{t-1}=1)\delta ^{t-1}+S_{T}^{D}(\gamma _{T-1}=1)\delta ^{T-1}\\ =\,&{\frac{\gamma _{0}}{2\left( 2-\gamma _{0}\right) }}+{\frac{1}{2}} \sum_{t=2}^{T-1}\delta ^{t-1}+{\frac{3}{2}}\delta ^{T-1}, \end{aligned} $$

  where γ₀ denotes the prior level of trust. In the final period, the principal will not pay the bonus, i.e. θ _T  = 0, because $S_{T}^{D}(\theta ^{T-1})\geq S_{T}^{H}(\theta ^{T-1})$ for any history of play. Since alternating strategies are ruled out by Eq. 16, any strategy that has θ₁ = 0, leads to a surplus of ${\mathcal{S}}(\theta _{1}=0,\theta _{2}=0,\ldots,\theta _{T}=0)={\frac{\gamma _{0}\left( 4-\gamma _{0}\right) }{2\left( 2-\gamma _{0})^{2}\right) }}<{\frac{3}{2}},$ which is clearly dominated by ${\mathcal{S}}(\varvec{ \theta }^{\ast })$ if r is sufficiently low (δ sufficiently close to 1). A strategy $\varvec{ \theta }^{\tau }=(\varvec{\theta}_{{\bf 1}}={\bf 1},\theta _{2}=1,\ldots,\theta _{\tau }=1,\theta _{\tau +1}=0,\ldots,\theta _{T}=0),2\leq \tau \leq T-1,$ leads to a surplus ${\mathcal{S}}(\varvec{ \theta }^{\tau })={\frac{\gamma _{0}}{2\left( 2-\gamma _{0})\right) }}+{\frac{1}{2}}\sum_{t=2}^{\tau }\delta ^{t-1}+{\frac{3}{2}}\delta ^{\tau }$. Hence, ${\mathcal{S}}(\varvec{ \theta }^{\ast })-{\mathcal{S}}( \varvec{ \theta }^{\tau })={\frac{1}{2}}\sum_{t=\tau +1}^{T-1}\delta ^{t-1}- {\frac{3}{2}}\left( \delta ^{\tau }-\delta ^{T-1}\right)$ is positive for all τ ≤ T −1 if r is sufficiently low.

• (b) If r is sufficiently large, only the first period payoff of the surplus function matters. The first period payoff is maximized by non-payment, θ₁ = 0. $\square$

Proof of Lemma 2

Assume r = 0. Consider a representative sequence $\theta ^{R}\left( \tau ^{F},1\right)$ consisting of τ^F payments and one non-payment. The surplus²⁷ from the second or higher repetition of the representative sequence $\theta ^{R}\left( \tau ^{F},1\right)$ is independent of the period in which the non-payment is placed within $\theta ^{R}\left( \tau ^{F},1\right)$. At each payment the agent recalls τ^F −1 payments and one non-payment, and in the period of non-payment the agent recalls τ^F payments (full trust). The surplus is equal to ${\mathcal{S}}=\tau ^{F}S^{H}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) +S^{D}\left( 1\right)$.²⁸

Now assume a second non-payment is optimal, d = 2. We consider two different strategies in placing the second non-payment. In strategy A it is placed immediately after the first non-payment, and in strategy B the two non-payments are not placed in a row. The surpluses related to strategies A and B are given by

$$ \begin{aligned} {\mathcal{S}}^{A}&=S^{D}\left( 1\right) +S^{D}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) +\left( \tau ^{F}-1\right) S^{H}\left( {\frac{\tau ^{F}-2}{\tau ^{F}}}\right) +S^{H}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) \\ {\mathcal{S}}^{B} &=2S^{D}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) +\left( \tau ^{F}-2\right) S^{H}\left( {\frac{\tau ^{F}-2}{\tau ^{F}}}\right) +2S^{H}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) \end{aligned} $$

The difference of surpluses is ${\mathcal{S}}^{A}-{\mathcal{S}}^{B}=S^{D}\left( 1\right) -S^{D}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) -\left( S^{H}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) -S^{H}\left( {\frac{\tau ^{F}-2}{\tau ^{F}}}\right) \right)$. Notice that $1-{\frac{\tau ^{F}-1}{\tau ^{F}}}={\frac{\tau ^{F}-1}{\tau ^{F}}}-{\frac{\tau ^{F}-2}{\tau ^{F}}}={\frac{1}{\tau ^{F}}}$. As both S ^D and S ^H are increasing convex functions of γ _(·), and as for the marginal surpluses it holds S ^D′ > S ^H′ for all $\gamma _{\left( \cdot \right) },{\mathcal{S}}^{A}-{\mathcal{S}}^{B}$ is strictly positive. Hence, if a second non-payment is optimal it must be placed immediately after the first non-payment. The same argumentation applies if d > 2 non-payments are optimal. Hence, if a repetition of representative sequences $\theta ^{R}\left( \tau ^{F},d\right)$ is consistent with equilibrium behavior, d non-payments must be placed in a row such that there is at most one change from payment to non-payment within a representative sequence. For positive but sufficiently low r the same result applies.$\square$

Proof of Proposition 4

Assume r = 0 for the whole proof. The proof consists of three steps:

• (a) We first prove that, independent of the initial trust γ₀ at the beginning of the first period, it is always optimal to establish full trust right from the beginning of the relationship by selecting τ^F payments in a row.

• (b) We next prove that given full trust has been established right from the beginning of the relationship at least one non-payment is optimal.

• (c) Finally, we show that repetition of the representative sequence is optimal.

(a) Assume T > 2τ^F. We first show that strategy $\varvec{ \theta }(\varvec{\tau}^{F},{\bf 0})=(\theta _{1}=\theta _{2}=\cdots=\theta _{\tau ^{F}}=1,0,0,\ldots,0),$ i.e., fulfill τ^F periods and then never again, dominates all strategies $\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i},{\bf 0})=(\theta _{1}=\theta _{2}=\cdots=\theta _{\tau ^{F}-i}=1,0,0,\ldots,0),\,i=1,\ldots,\tau ^{F}-1$ if γ₀ < 1. Strategy $\varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0})$ yields the surplus²⁹

$$ {\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0} ))=\sum_{t=1}^{\tau ^{F}}S^{H}(\theta ^{t-1})+\sum_{t=\tau ^{F}+1}^{2\tau ^{F}}S^{D}(\theta ^{t-1}), $$

(18)

whereas strategy’s $\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i},{\bf 0} )$ surplus obtains as

$$ {\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i},{\bf 0} ))=\sum_{t=1}^{\tau ^{F}-i}S^{H}(\theta _{i}^{t-1})+\sum_{t=\tau ^{F}-i}^{2\tau ^{F}-i}S^{D}(\theta _{i}^{t-1}), $$

(19)

where the subscript i at θ ^t−1 _i indicates the history under $\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i},{\bf 0})$ compared with $\varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0})$. Note that Eq. 18 contains i strictly positive elements more than Eq. 19 due to i additional payments and limited recall. The profit difference $\Delta ={\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F}, {\bf 0}))-{\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -1,0}))$ using Eq. 18 and 19 amounts to

$$ \Delta =S^{H}(\theta ^{\tau ^{F}-1})+\sum_{t=\tau ^{F}+1}^{2\tau ^{F}}\left[ S^{D}(\theta ^{t-1})-S^{D}(\theta _{i}^{t-2})\right]. $$

Because of ${\frac{dS^{H}}{d\gamma _{t-1}}}>0$ and ${\frac{dS^{D}}{d\gamma _{t-1}}}>0$ for any history, payment in period τ^F under ${\mathcal{S}}( \varvec{ \theta }(\varvec{ \tau}^{F}{\bf,0}))$ provides for

$$ S^{D}(\theta ^{t-1})-S^{D}(\theta _{i}^{t-1})>0,\hbox { }t=\tau ^{F}+1,\ldots,2\tau ^{F}. $$

(20)

This proves ${\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0}))> {\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -1},{\bf 0})).$ By iteration, it can be shown that ${\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau} ^{F}{\bf -i},{\bf 0}))>{\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F} {\bf -i-1},{\bf 0})),\,i=2,\ldots,\tau ^{F}-1.$ Thus, ${\mathcal{S}}( \varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0}))>{\mathcal{S}}(\varvec{ \theta } (\varvec{ \tau}^{F}{\bf -i},{\bf 0}))$ for all i = 1, …, τ^F − 1.

Next, we show that strategy $\varvec{ \theta }(\varvec{ \tau}^{F},\theta _{\tau ^{F}+1},\ldots,\theta _{T}),$ where the θ _t ’s, t = τ^F + 1, …, T, are optimally chosen, dominates all other possible strategies. Assume that strategy $\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i},{\bf 0})$ is changed by replacing a non-payment in period $t=\widetilde{t}=\tau ^{F}-i+1{,}\ldots{,}(2\tau ^{F}-i)$ with a payment. Because ${\frac{d^{2}S^{H}}{ d\gamma _{t-1}^{2}}}>0$ and ${\frac{d^{2}S^{D}}{d\gamma _{t-1}^{2}}}>0$ —which only holds in case of limited recall so that the sample size or the denominator in the expected level of trust remains constant at τ^F—marginal gains (losses) from payment (non-payment) are increasing in previous payments (non-payments). Hence, if a non-payment is replaced by a payment it has to be in period $\widetilde{t}=\tau ^{F}-i+1.$ Optimality of that replacement follows from the steps of the proof above implying ${\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i+1}, {\bf 0}))>{\mathcal{S}}(\varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i}, {\bf 0})).$ Again, by iteration the optimality of additional replacements in periods $\widetilde{t}=\tau ^{F}-i+2,\ldots,\tau ^{F}$ can be shown leading (again) to $\varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0})\succ \varvec{ \theta }(\varvec{ \tau}^{F}{\bf -i},{\bf 0})$ for all i = 1,…,τ^F − 1. Obviously, $\varvec{ \theta }(\varvec{ \tau}^{F},\theta _{\tau ^{F}+1},\ldots,\theta _{T})\succeq \varvec{ \theta }(\varvec{ \tau}^{F},{\bf 0} )$ proving optimality of $\varvec{ \theta }(\varvec{ \tau}^{F},\theta _{\tau ^{F}+1},\ldots,\theta _{T}).$

(b) Assume a sequence of τ^F payments has been selected leading to full trust. Now compare the following sequences

$$ \varvec{ \theta }_{d=1} =(\theta _{\tau ^{F}+1}=0,1,1,\ldots,\theta _{2\tau ^{F}+2}=1) $$

(21)

$$ \varvec{ \theta }_{d=0} =(\theta _{\tau ^{F}+1}=1,1,1,\ldots,\theta _{2\tau ^{F}+2}=1), $$

(22)

starting in period τ^F + 1. Note that both sequences consist of (τ^F + 1) elements to ensure that full trust is (re)established at the end of the sequences. The crucial step in the proof is that the trust level given $\varvec{ {\theta}}_{d=1}$ remains constant after decision in period $\tau ^{F} + 1,\theta _{{\tau ^{F} + 1}} = 0$, because when moving on from period (τ^F + 1 + i) to (τ^F + 1 + i + 1), i ∈ {1, 2,…,τ^F}, the agent’s limited recall capability “deletes” the payment in period (i + 1) from memory while “storing” the payment from period (τ^F + 1 + i). As such the single non-payment in period τ^F + 1 reduces the trust level from full trust 1 under $\varvec{ \theta }_{d=1}$ to ${\frac{\tau ^{F}-1}{ \tau ^{F}}}$. Surpluses associated with Eqs. 21 and 22 then obtain as

$$ \begin{aligned} {\mathcal{S}}(\varvec{ \theta }_{d=1}) =\,&{\frac{3}{2}}+\tau ^{F}\cdot S^{H}\left( {\frac{\tau ^{F}-1}{\tau ^{F}}}\right) \\ =\,&{\frac{3}{2}}+\tau ^{F}\cdot {\frac{\tau ^{F}-1}{2(\tau ^{F}+1)}} \\ {\mathcal{S}}(\varvec{ \theta }_{d=0}) =\,&(\tau ^{F}+1)\cdot {\frac{1}{2}}. \end{aligned} $$

Some algebra shows that ${\mathcal{S}}(\varvec{ \theta }_{d=1})>{\mathcal{S}}( \varvec{ \theta }_{d=0})$ holds for any τ^F.

(c) We know from part (a) and (b) that the principal’s optimal strategy exhibits payment from period one to τ^F and at least one non-payment thereafter. From lemma 2 it is known that switching back and forth between one payment and one non-payment will never be optimal. Hence, the first sequence that is played consists of τ^F payments followed by d ≥ 1 non-payments, $\theta ^{R}(\tau ^{F},d)=\left( \theta _{1}=1,\theta _{2}=1,\ldots,\theta _{\tau ^{F}}=1,\theta _{\tau ^{F}+1}=0,\ldots,\theta _{\tau ^{F}+d}=0\right)$. After $\theta ^{R}(\tau ^{F},d)$ has been played once, induced trust is again less than one. Applying the same arguments as in (a) and (b) it is again optimal to re-induce full trust by paying the bonus τ^F-times in a row and then harvesting it by d non-payments in a row. Hence, at the optimum the representative sequence θ^R(τ^F,d) will be repeated as long as possible given there are at least τ^F-periods remaining to harvest trust after it has been raised to its maximum. The periods after the last repetition of θ^R(τ^F,d) are subject to separate optimization.

As the results in (a), (b), and (c) are derived for r = 0 they also hold for a sufficiently low interest rate.$\square$

Proof of Proposition 5

The idea of the proof is to transform strategies into profit annuities (the optimal strategy will have the highest profit annuity). Optimal strategies are characterized by their representative sequence. Effects of the ex ante distribution of trust are eliminated after initial play of τ^F payments (which have already been proven optimal). Subsequent repetitions of the representative sequence will then be played with recurring levels of trust solely determined by τ^F and d. The first decision and its associated profit which is not influenced by ex ante trust is the first non-payment after τ^F payments. Therefore we rearrange the representative sequence such that d non-payments are followed by τ^F payments. With τ^F and d given, the profit annuity a(d,τ^F) based on the decision sequence θ(d,τ^F) obtains as

$$ a(d,\tau ^{F})=AF(d,\tau ^{F})\cdot \pi _{0}(d,\tau ^{F}), $$

(23)

where $\pi _{0}=\left( \sum_{t=1}^{d}{\frac{S_{t}^{D}(\gamma _{t-1})}{ (1+r)^{t}}}+\sum_{t=d+1}^{\tau ^{F}+d}{\frac{S_{t}^{H}(\gamma _{t-1})}{ (1+r)^{t}}}\right)$ denotes the present value resulting from playing the sequence once, and $AF(d,\tau ^{F})={\frac{(1+r)^{(\tau ^{F}+d)}\cdot r}{ (1+r)^{(\tau ^{F}+d)}-1}}$ is the annuity factor. (The dependence of a(d,τ^F) on r is suppressed for notational brevity. For the same reason, both d and τ^F are assumed to be continuous.)

Lemma 3

For a given τ ^F, the profit annuity a(d,τ) has a unique maximizer d ^*.

Proof of Lemma 3

Note ${\frac{\partial }{\partial d}}\,AF(d,\tau ^{F})<0.$ Furthermore, the first term $\sum_{t=1}^{d}{\frac{S_{t}^{D}(\gamma _{t-1})}{(1+r)^{t}}}$ of π ₀(d,τ^F) is increasing in d: adding one non-payment adds ${\frac{ S_{d+1}^{D}(\gamma _{d})}{(1+r)^{d+1}}}$ to the sum while leaving previous summands unchanged; the second term $\sum_{t=d+1}^{\tau ^{F}+d}{\frac{ S_{t}^{H}(\gamma _{t-1})}{(1+r)^{t}}}$ is decreasing in d: adding one non-payment decreases all S ^H _t (γ _t−1) because induced trust in all periods decreases. Hence either

• (a) ${\frac{\partial }{\partial d}}\pi _{0}(d,\tau ^{F}) <0\hbox { } \forall d,\hbox { or}$

• (b) ${\frac{\partial }{\partial d}}\pi _{0}(d,\tau ^{F}) >0\hbox { } \forall d,\hbox { or}$

• (c) ${\frac{\partial }{\partial d}}\pi _{0}(d,\tau ^{F}) <0\hbox { } \forall d\in \lbrack 1,\overline{d}] \hbox { and }{\frac{\partial }{\partial d}}\pi _{0}(d,\tau ^{F}) >0\hbox { if }d\in ( \overline{d},\tau ^{F}-1],\hbox { or}$

• (d) ${\frac{\partial }{\partial d}}\pi _{0}(d,\tau ^{F}) > 0\hbox { } \forall d\in \lbrack 1,\widetilde{d}] \hbox { and }{\frac{\partial }{\partial d}}\pi _{0}(d,\tau ^{F}) < 0\hbox { if }d\in ( \widetilde{d},\tau ^{F}-1]$

must hold. The proof of proposition 4 shows a(1, τ^F) > a(0, τ^F) for all τ^F. Hence, either (b) or (d) holds. It follows, either (i) ${\frac{\partial }{\partial d}}a(d,\tau ^{F})>0$ for $\forall d\leq \left( \tau ^{F}-1\right)$ holds, or (ii) ${\frac{\partial }{\partial d}}a(d,\tau ^{F})>0$ for $\forall d\in \lbrack 1,d^{\ast }],d^{\ast }\leq \widetilde{d},$ and ${\frac{\partial }{\partial d}}a(d,\tau ^{F})<0$ if $d\in (d^{\ast },\tau ^{F}-1]$ must hold. Therefore $d^{\ast }\leq \left( \tau ^{F}-1\right)$ is the unique maximizer of a(d, τ^F). $\square$

Now assume $d^{\ast }=(\tau ^{F}-1)$ is the unique maximizer of a(d, τ^F). Then it is obvious that d increases in τ^F. Assume contrary to it $d^{\ast }<\left( \tau ^{F}-1\right)$ to be optimal; hence $\left( d^{\ast }+1\right)$ is not optimal given τ^F. The condition for optimality of (d ^* + 1) using Eq. 23 obtains as

$$ \begin{aligned} a(d^{\ast }+1,\tau ^{F}) >&a(d^{\ast },\tau ^{F})\\ {\frac{AF(d^{\ast }+1,\tau ^{F})}{AF(d^{\ast },\tau ^{F})}}\cdot \pi _{0}(d^{\ast }+1,\tau ^{F}) >&\pi _{0}(d^{\ast },\tau ^{F}) \end{aligned} $$

(24)

Observe, $\lim_{\tau ^{F}\rightarrow \infty }{\frac{AF(d^{\ast }+1,\tau ^{F}) }{AF(d^{\ast },\tau ^{F})}}=1.$ Therefore the left-hand side of (24) converges to $\lim_{\tau ^{F}\rightarrow \infty }\pi _{0}(d^{\ast }+1,\tau ^{F})=\pi _{0}(d^{\ast },\tau ^{F})+{\frac{ S_{d^{\ast }+1}^{D}(\gamma _{d^{\ast }})-S_{d^{\ast }+1}^{H}(\gamma _{d^{\ast }})}{(1+r)^{d^{\ast }+1}}}$ and the relation in (24) will hold if τ^F increases sufficiently.$\square$