A Behavioral Model on the Effects of Information Sharing on Supply Chain Performance

Abstract

This section investigates the impact of information sharing in a behavioral model assuming that a certain fraction of buyers give honest reports, and the supplier reacts to these reports by adjusting his beliefs according to Bayes’ rule. First, Sect. 6.1 briefly summarizes the principal–agent literature assuming that not all agents (here: buyers) use their private information entirely strategically, while showing that this assumption is supported by experimental results. Afterwards, Sect. 6.2 depicts how communication, trust, and trustworthiness can be formalized in the strategic lotsizing framework. Then, Sect. 6.3 evaluates the impact of information sharing assuming that the deceptive buyer gives his signals without considering his actual cost position, while Sect. 6.4 discusses the impact of strategic reporting. Finally, Sect. 6.5 summarizes the results and gives some managerial insights.

Appendix: Proof of Theorems

Theorem 1

The suppliers expected costs do not increase due to the adjustment of the a priori distribution as long as $\hat{\alpha } \le {\min_i}\left[ {\frac{{{{\hat{\phi }}_i} \cdot \alpha }}{{{{\hat{\phi }}_i} \cdot \alpha + {\phi_i} - {\phi_i}\alpha }}} \right],i = 1,...,n$ holds.

Proof

In the following it is shown that the supplier estimates the actual a posteriori distribution ${p_i}({h_i}|{S_i})$ more accurately if the condition in Theorem 1 holds.

(a) Adjustment of the probability that corresponds to the signal: ${\hat{p}_i}({h_i}|{S_i}),\forall i = 1,...,n$

First, it is shown that the perceived a posteriori probability is always lower than the actual a posteriori probability if $\hat{\alpha } \le \frac{{{{\hat{\phi }}_i}\alpha }}{{{{\hat{\phi }}_i}\alpha + {\phi_i} - {\phi_i}\alpha }}$ holds.

$$ \begin{array}{lll}{{\hat{p}}_i}({h_i}|{S_i}) = \frac{{\hat{\alpha }{p_i} + (1 - \hat{\alpha }){p_i}{{\hat{\phi }}_i}}}{{\hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i}}} \le \frac{{\alpha {p_i} + (1 - \alpha ){p_i}{\phi_i}}}{{\alpha {p_i} + \left( {1 - \alpha } \right){\phi_i}}} = {p_i}({h_i}|{S_i}) \hfill \\ \to - \frac{{{p_i}\left( { - \alpha {{\hat{\phi }}_i} + \alpha {{\hat{\phi }}_i}\hat{\alpha } - {\phi_i}\hat{\alpha }{p_i} + {\phi_i}\alpha \hat{\alpha }{p_i} + \hat{\alpha }{\phi_i} - \hat{\alpha }{\phi_i}\alpha + {{\hat{\phi }}_i}\alpha {p_i} - {{\hat{\phi }}_i}\hat{\alpha }\alpha {p_i}} \right)}}{{\left( {\alpha {p_i} + \left( {1 - \alpha } \right){\phi_i}} \right)\left( {\hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i}} \right)}} \ge 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots \hfill \\ \to \hat{\alpha } \le \frac{{{{\hat{\phi }}_i}\alpha }}{{{{\hat{\phi }}_i}\alpha + {\phi_i} - {\phi_i}\alpha }} \hfill \\ \end{array} $$

As the supplier’s estimation needs to be more accurately for every signal ${S_i},i = 1,...,n$ (note that there is no adjustment for ${S_{n + 1}}$), it follows:

$$ \hat{\alpha } \le {\min_i}\left[ {\frac{{{{\hat{\phi }}_i} \cdot \alpha }}{{{{\hat{\phi }}_i} \cdot \alpha + {\phi_i} - {\phi_i}\alpha }}} \right],i = 1,...,n. $$

Second, it needs to be shown that ${\hat{p}_i}({h_i}|{S_i}) \ge {p_i}$ holds:

$$ \begin{array}{lll} {p_i} \le \frac{{\hat{\alpha }{p_i} + (1 - \hat{\alpha }){p_i}{{\hat{\phi }}_i}}}{{\hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i}}} \hfill \\ \hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i} \le \hat{\alpha } + (1 - \hat{\alpha }){{\hat{\phi }}_i} \hfill \\ \to {p_i} \le 1\; \hfill \\ \end{array} $$

Hence, it follows that

$$ fbox{p_i} \le {\hat{p}_i}({h_i}|{S_i}) \le {p_i}({h_i}|{S_i}),\;\forall i = 1,...,n. $$

(b) Adjustment of probabilities that do not correspond to the signal, i.e., ${\hat{p}_k}({h_k}|{S_i}),\forall k = 1,...,n;i = 1,...,n + 1;i \ne k$

$$ \begin{array}{lll} {{\hat{p}}_k}({h_k}|{S_i}) = \frac{{(1 - \hat{\alpha }){p_k}{{\hat{\phi }}_i}}}{{\hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i}}} \ge \frac{{(1 - \alpha ){p_k}{\phi_i}}}{{\alpha {p_i} + \left( {1 - \alpha } \right){\phi_i}}} = {p_k}({h_k}|{S_i}) \hfill \\ \to - \frac{{{p_k}{p_i}( - \hat{\alpha }{\phi_i} + \hat{\alpha }{\phi_i}\alpha + \alpha {{\hat{\phi }}_i} - \alpha \hat{\alpha }{{\hat{\phi }}_i})}}{{\left( {\alpha {p_i} + \left( {1 - \alpha } \right){\phi_i}} \right)\left( {\hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i}} \right)}} \le 0 \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ \to \hat{\alpha } \le \frac{{{{\hat{\phi }}_i}\alpha }}{{{{\hat{\phi }}_i}\alpha + {\phi_i} - {\phi_i}\alpha }} \hfill \\ \end{array} $$

Hence, it follows:

$$ \hat{\alpha } \le {\min_i}\left[ {\frac{{{{\hat{\phi }}_i} \cdot \alpha }}{{{{\hat{\phi }}_i} \cdot \alpha + {\phi_i} - {\phi_i}\alpha }}} \right],i = 1,...,n $$

Furthermore, it needs to be shown that ${\hat{p}_k}({h_k}|{S_i}) \le {p_k}$ holds:

$$ \begin{array}{lll} {p_k} \ge \frac{{(1 - \hat{\alpha }){p_k}{{\hat{\phi }}_i}}}{{\hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right){{\hat{\phi }}_i}}} \hfill \\ \hat{\alpha }{p_i} + \left( {1 - \hat{\alpha }} \right)\hat{\phi } \ge (1 - \hat{\alpha }){{\hat{\phi }}_i} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ \to {p_i} \ge 0 \hfill \\ \end{array} $$

Hence it follows that

$$ {p_k}({h_k}|{S_i}) \leqslant {\hat{p}_k}({h_k}|{S_i}) \leqslant {p_k},\;\forall i = 1,...,n + 1;k = 1,...,n;i \ne k $$

The same argumentation follows for $\hat{\alpha } \ge {\min_i}\left[ {\frac{{{{\hat{\phi }}_i} \cdot \alpha }}{{{{\hat{\phi }}_i} \cdot \alpha + {\phi_i} - {\phi_i}\alpha }}} \right],i = 1,...,n$:

$$ \begin{array}{lll}{p_i} \le {p_i}({h_i}|{S_i}) \le {{\hat{p}}_i}({h_i}|{S_i})\;\forall i = 1,...,n \hfill \\{{\hat{p}}_k}({h_k}|{S_i}) \le {p_k}({h_k}|{S_i}) \le {p_k},\;\forall i = 1,...,n + 1;k = 1,...,n;i \ne k \hfill \\ \end{array} $$

In this case, however, the change of the supplier’s expected costs depends on the specific cost structure.

Theorem 2.1

The honest buyer’s expected costs increase due to truthful signaling.

The informational rent denotes the costs savings that occur for the buyer in comparison to the outside option, i.e., the alternative supplier. Let $I{R_i}$ denote informational rent of the buyer who faces holding costs of ${h_i}$. From condition 2 (see Chap. 3) it follows:

$$ \begin{array}{lll} \frac{{{h_n}}}{2}{q_n} - {Z_n} = {C_{AS}} \hfill \\ \to I{R_n} = {C_{AS}} + {Z_n} - \frac{{{h_n}}}{2}{q_n} = 0 \hfill \\ \frac{{{h_{n - 1}}}}{2}{q_{n - 1}} - {Z_{n - 1}} = \frac{{{h_{n - 1}}}}{2}{q_n} - {Z_n} \hfill \\ \to I{R_{n - 1}} = {C_{AS}} + {Z_{n - 1}} - \frac{{{h_{n - 1}}}}{2}{q_{n - 1}} =... = I{R_n} + \frac{{{q_n}}}{2}\left( {{h_n} - {h_{n - 1}}} \right) > 0,\;\rm as\;{h_{n - 1}} < {h_n} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; \vdots \hfill \\ I{R_i} =... = {C_{AS}} + {Z_{i + 1}} - \frac{{{h_i}}}{2}{q_{i + 1}} = I{R_{i + 1}} + \frac{{{q_{i + 1}}}}{2}\left( {{h_{i + 1}} - {h_i}} \right) \hfill \\ \end{array} $$

Hence, if all ${q_{i + 1}}^k,...,{q_{n - 1}}^k,{q_n}^k$ decrease given a signal ${S_k}$, then the informational rent $I{R_i}$ decreases as well.

The impact on the order sizes ${q_m}^k,m = i + 1,...,n$ is analyzed by computing the change of ${\gamma_m}^k$ given the changes of ${\hat{p}_t}({h_t}|{S_k}),t = i + 1,...,n$. From $\frac{{\partial {\gamma_m}^k}}{{\partial {{\hat{p}}_t}({h_t}|{S_k})}} \le 0,\forall t = i + 1,...,n$ it follows directly that ${q_m}^k \le {q_m}^{AI}$ as long as ${p_t} \ge {\hat{p}_t}({h_t}|{S_k})$. The condition ${p_t} \ge {\hat{p}_t}({h_t}|{S_k})$ holds for all $H \le \tilde{h}$ (see Theorem 1), i.e., as long as the buyer reports truthfully or understates his holding costs.

Theorem 2.2

The expected costs of the deceptive buyer can either increase or decrease due to communication.

The deceptive buyer may either (a) report accidentally truthful, i.e., $S = \hat{h}$, or (b) misreport his holding cost, i.e., $S \ne \hat{h}$.

Case (a): Accidental Truthful Reporting

$S = \hat{h}$ may occur if ${\phi_i} > 0$ holds.

From Theorem 2.1 it follows directly that the deceptive buyer is worse off if he reports truthfully.

Case (b): Deceptive Reporting

Deceptive reporting is formalized by $S \ne \hat{h}$. The case of the deceptive reporting is divided into three subclasses, (b1) an understatement of holding cost, (b2) an overstatement of holding cost, but not to the maximum extent and (b3) an overstatement to the maximum extent.

Case (b1): $S < \hat{h}$ (understatement of holding costs)

From Theorem 2.1 it follows that the deceptive buyer can only be worse off if he understates his holding cost.

Case (b2): $\hat{h} < S < {h_n}$ (overstatement, not to the maximum extent)

In this case, the deceptive buyer can either be better off or worse off. If the reduction of the informational rents $I{R_k},....,I{R_n}$ given ${S_k}$ and $\hat{h} = {h_i}$ is compensated by the increase of the informational rents $I{R_i},....,I{R_{k - 1}}$, then he is better off. Otherwise, he is not. However, this depends on the specific cost structure.

Case (b3): ${h_n} = S \ge \hat{h}$ (overstatement to the maximum extent)

From Theorem 2.1 and condition (1) (see Chap. 3) it follows directly that the deceptive buyer can only be better off if he constantly signals ${S_n}$. In this case, the supplier adjusts the order quantity ${q_n}^n$ upwards, i.e., ${q_n}^n > {q_n}^{AI}$. This leads to an increase of all informational rents $I{R_i},i = 1,...,n - 1$.

Theorem 3.2

As long as $\alpha \ge \frac{{\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } \,+\, \sum\limits_i {{p_i}{\phi_i}\Delta C{D_i}^i} }}{{\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } \,-\, \sum\limits_i {\left( {1 - {\phi_i}} \right){p_i}\Delta C{D_i}^i} }} = {\alpha_{crit}}\left( {\hat{\alpha },{{\hat{\phi }}_i}} \right)$ holds, communication enhances the supply chain performance, and vice versa.

$$ \begin{array}{lll}\Delta CD = \left( {1 - \alpha } \right)\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } + \alpha \sum\limits_i {{p_i}\Delta C{D_i}^i} \hfill \\ \quad+ \left( {1 - \alpha } \right)\sum\limits_i {{p_i}{\phi_i}\Delta C{D_i}^i} \le 0 \hfill \\ \hskip6pc\vdots \hfill \\ \sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta CD_i^k} } + \sum\limits_i {{p_i}{\phi_i}\Delta CD_i^i} \hfill \\ \hskip5.7pc\le \hfill \\ \alpha \sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta CD_i^k} } - \alpha \sum\limits_i {\left( {1 - {\phi_i}} \right){p_i}\Delta CD_i^i} \hfill \\ \hskip6.2pc\vdots \hfill \\ \alpha \ge \frac{{\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta CD_i^k} } + \sum\limits_i {{p_i}{\phi_i}\Delta CD_i^i} }}{{\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta CD_i^k} } - \sum\limits_i {\left( {1 - {\phi_i}} \right){p_i}\Delta CD_i^i} }} = {\alpha_{crit}}\left( {\hat{\alpha },{{\hat{\phi }}_i}} \right) \hfill \\ \end{array} $$

As $\Delta C{D_i}^i \le 0$ it can be easily shown that ${\alpha_{crit}} \le 1$ holds.

Theorem 3.3

The range of levels of trustworthiness for which communication is an appropriate coordination mechanism decreases with increasing levels of supplier’s trust.

Let ${\alpha_{\min }}$ denote the level of trustworthiness for which the coordination deficit reaches its minimum for a given a certain level of trust ${\hat{\alpha }_{\min }}$.

${\alpha_{\min }}$ follows from:

$$ \begin{array}{lll} \Delta CD = \left( {1 - \alpha } \right)\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } + \alpha \sum\limits_i {{p_i}\Delta CD_i^i} \hfill \\ + \left( {1 - \alpha } \right)\sum\limits_i {{p_i}{\phi_i}\Delta C{D_i}^i} = 0 \hfill \\\to \hfill \\\frac{{\partial \Delta CD}}{{\partial \hat{\alpha }}} = \left( {1 - \alpha } \right)\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\frac{{\partial \Delta C{D_i}^k}}{{\partial \hat{\alpha }}}} } + \alpha \sum\limits_i {{p_i}\frac{{\partial \Delta C{D_i}^i}}{{\partial \hat{\alpha }}}} \hfill \\ + \left( {1 - \alpha } \right)\sum\limits_i {{p_i}{\phi_i}\frac{{\partial \Delta C{D_i}^i}}{{\partial \hat{\alpha }}}} = 0 \hfill \\\to \hfill \\{\alpha_{\min }} = \frac{{\sum\limits_i {{p_i}{\phi_i}} {{\left. {\displaystyle\frac{{\partial \Delta C{D_i}^i}}{{\partial \hat{\alpha }}}} \right|}_{\hat{\alpha } = {{\hat{\alpha }}_{\min }}}} + \sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}{{\left. {\displaystyle\frac{{\partial \Delta C{D_i}^k}}{{\partial \hat{\alpha }}}} \right|}_{\hat{\alpha } = {{\hat{\alpha }}_{\min }}}}} } }}{{{{\left. {\sum\limits_i}P i (\phi_i-1){\displaystyle\frac{{\partial \Delta C{D_i}^i}}{{\partial \hat{\alpha }}}} \right|}_{\hat{\alpha } = {{\hat{\alpha }}_{\min }}}} + \sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}{{\left. {\displaystyle\frac{{\partial \Delta C{D_i}^k}}{{\partial \hat{\alpha }}}} \right|}_{\hat{\alpha } = {{\hat{\alpha }}_{\min }}}}} } }} < 1 \hfill \\\end{array} $$

It follows that $\frac{{\partial \Delta CD}}{{\partial \hat{\alpha }}} \le 0$, if $\alpha \ge {\alpha_{\min }}$, and vice versa.

Additionally, $\frac{{\partial \Delta CD}}{{\partial \alpha }} \le 0$ holds:

$$ \begin{array}{lll} \Delta CD = \left( {1 - \alpha } \right)\sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } + \alpha \sum\limits_i {{p_i}\Delta C{D_i}^i} \hfill \\ \hskip2.7pc+ \left( {1 - \alpha } \right)\sum\limits_i {{p_i}{\phi_i}\Delta C{D_i}^i} \hfill \\ \frac{{\partial \Delta CD}}{{\partial \alpha }} = - \sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } + \sum\limits_i {{p_i}\Delta C{D_i}^i} \hfill \\ \sum\limits_i {{p_i}{\phi_i}\Delta C{D_i}^i} \le 0 \hfill \\ \vdots \hfill \\ \sum\limits_i {{p_i}\left( {1 - {\phi_i}} \right)\Delta C{D_i}^i} \le \sum\limits_i {\sum\limits_{k,i \ne k} {{p_i}{\phi_k}\Delta C{D_i}^k} } \hfill \\ \end{array} $$

As $\Delta C{D_i}^i \le 0\;\forall i = 1,...,n$ and $\Delta C{D_i}^k \ge 0,\forall i,k = 1,...,n;i \ne k$ (see Theorem 2.1) it follows directly that $\frac{{\partial \Delta CD}}{{\partial \alpha }} \le 0$ is true.

If $\alpha = {\alpha_{crit}}$ holds, it follows that $\Delta CD = 0$ (see Theorem 3.2). Hence, it follows that $\alpha < {\alpha_{crit}}\; \to \;\Delta CD > 0\;{\rm and}\;\alpha > {\alpha_{crit}}\; \to \;\Delta CD < 0,\;{\rm respectively}.$

Hence, it follows directly that ${\alpha_{\min }} > {\alpha_{crit}}$. Since $\frac{{\partial \Delta CD}}{{\partial \hat{\alpha }}} \ge 0$ if ${\alpha_{\min }} \ge \alpha$, it follows directly that $\frac{{\partial {\alpha_{crit}}}}{{\partial \hat{\alpha }}} \ge 0$.