Principal/Two-Agent model with internal signal

Abstract

Our novel game-theoretic Principal/Two-Agent model ensures that the Principal has a reliable internal signal about the Agents’ invested work and effort. Analysing the dominant strategies deductively proves that suboptimal results cannot be prevented with focus on evaluation, implying that quality of work, not its evaluation, must be considered as the most important outcome of the process. The objective of the paper is to establish a new game theoretic model that can be used as a tool for policy makers and managers to motivate Agents and ensure high quality results. Additionally, the model can be used to determine the awards in a company bonus system. The newly developed model is an extended Principal-Agent model with an internal game between two Agents, whose payoff structure can be set in order to ensure truthful implementation of the internal signal to the Principal. By analysing the dominant strategies, we determined the conditions that ensure each Nash equilibrium of the game to manifest the desired outcome. The model presents a novel approach to alignment of interests, for example, in economy, social problems (e.g. policy making for educational process), management, project management, and the mentor-apprentice relationship.

Appendix: Proof of Theorem 1

Lemma 1

If for each $x\in \lbrace A,B \rbrace$, $df_x$ and $dg_x$ are strictly positive numbers and that $e_x>o_x>u_x>i_x$ , then the following is true:

The pairs of strategies $( \lbrace C\; H, C\; L \rbrace , \lbrace C\; \cdot \rbrace )$ are pure-strategy Nash equilibria of the Principal/Two-Agent model with internal signal, if and only if the following conditions hold:

$$\begin{aligned}&deo_A \ge df_A; \end{aligned}$$

(5)

$$\begin{aligned}&deu_B \ge df_B. \end{aligned}$$

(6)

Proof

First, we assume that (i, j) is a pure-strategy Nash equilibrium for each $i \in \{1,2\}$ and $j \in \{ 1,\ldots ,8\}$, i.e. all pairs resulting in the desired outcome I.

• As the strategy (1, 2) is a pure-strategy Nash equilibrium, it dominates the strategy (5,2). Therefore, $\pi _A(1,2)\ge \pi _A(5,2)$, expanded $f'_A+g_A+e_A \ge f_A+g_A+o_A$, implying (5).

• As the strategy (2, 1) is a pure-strategy Nash equilibrium, it dominates the strategy (2, 9). Therefore, $\pi _B(2,1)\ge \pi _B(2,9)$, expanded $f_B'+g_B+e_B \ge f_B+g_B+u_B$, implying (6).

For the converse implication, we assume (5) and (6). For each (i, j), $i\in \{1,2\}$, $j\in \{ 1,\ \ldots ,\ 8 \}$, and for each $k\in \{1,\ \ldots ,\ 8\},$ we need to prove

$$\begin{aligned} \pi _A(i,j)\ge \pi _A(k,j), \end{aligned}$$

(7)

and, for each $l \in \{ 1,\ \ldots ,\ 16\},$ also

$$\begin{aligned} \pi _B(i,j)\ge \pi _B(i,l). \end{aligned}$$

(8)

• 1. We prove (1) as follows:

  • (a) The equality $\pi _A(i,j)= \pi _A (k,j)$ holds for $i,k \in \{ 1,2\}$, and $j\in \{1,\ \ldots ,\ 8\}.$

  • (b) (5) implies $f_A'+g_A+e_A> f_A+g_A+o_A$ and therefore $\pi _A(i,j)> \pi _A(k,j)$ for $i\in \{1,2\}$, $j \in \{2\}$, and $k \in \{5,6\}$.

  • (c) $dg_A> 0$ implies $f_A'+g_A+e_A> f_A'+g_A'+e_A$, therefore $\pi _A(i,j)> \pi _A(k,j)$ for $i\in \{1,2\}$, $j \in \{1,\ \ldots ,\ 8\}$, and $k \in \{3,4\}$.

    • i. $dg_A>0$ with (5) implies $f_A'+g_A+e_A>f_A+g_A'+o_A$, therefore $\pi _A(i,j)> \pi _A(k,j)$ for $i\in \{1,2\}$, and for $j \in \{1,3,4\}$ and $k \in \{5,\ \ldots ,\ 8\}$, as well as for $j \in \{2\}$ and $k\in \{7,8\}$.

    • ii. $e>o>u>i$ implies $dei_A> deo_A$. $dei_A> deo_A$ with (5) implies $f_A'+g_A+e_A>f_A+g_A+i_A$, therefore $\pi _A(i,j)> \pi _A(k,j)$ for $i\in \{1,2\}$, $j \in \{6\}$, and $k \in \{6,8\}$.

      • A. $dg_A>0$ and $dei_A> deo_A$ with (5) implying $f_A'+g_A+e_A>f_A+g_A'+i_A$, therefore $\pi _A(i,j)> \pi _A(k,j)$ for $i\in \{1,2\}$ and for $j \in \{6\}$ and $k \in \{5,7\}$; as well as for $j \in \{5,7,8\}$ and $k \in \{5,\ \ldots ,\ 8\}$.

• 2. We prove (8) as follows.

  • (a) The equality $\pi _B(i,j)= \pi _B (i,l)$ holds for $i \in \{ 1,2\}$ and $j,l \in \{1,\ \ldots ,\ 8\}.$

  • (b) (6) implies $f_B'+g_B+e_B>f_B+g_B+u_B$, therefore $\pi _B(i,j)> \pi _B (i,l)$ for $i \in \{ 2\}$, $j\in \{1,\ \ldots ,\ 8\}$, and $l\in \{ 9,\ \ldots ,\ 16 \}$.

    • i. $dg_B>0$ with (6) implies $f'_B+g_B+e_B>f_B+g_B'+u_B$, therefore $\pi _B(i,j)> \pi _B (i,l)$ for $i \in \{ 1\}$, $j\in \{1,\ \ldots ,\ 8\}$, and $l\in \{ 9,\ \ldots ,\ 16 \}$.

Since the strategies (i, j) for $i\in \{1,2\}$ and $j \in \{1,\ \ldots ,\ 8\}$ dominate respective rows and columns, and have equal utilities by themselves, each of these pairs is a pure-strategy Nash equilibrium. Table 5 summarizes the applied arguments. $\square$

Table 5 Arguments for strategies being the pure-strategy Nash equilibria dominating respective strategies

In Theorem 1, we identify sufficient and necessary conditions for these pairs to be the only Nash equilibria, hence eliminating any rational desire for other pairs of strategies to become stable behaviour.

Theorem 1

If, for each $x\in \lbrace A,B \rbrace$, $df_x$ and $dg_x$ are strictly positive numbers and $e_x>o_x>u_x>i_x$ , then the following is true:

The pairs of strategies $( \lbrace C\; H, C\; L \rbrace , \lbrace C\; \cdot \rbrace )$ are the only pure strategy Nash equilibria of the Principal/Two-Agent model with internal signal, if and only if all of the following conditions hold:

$$\begin{aligned}&deo_A \ge df_A; \end{aligned}$$

(9)

$$\begin{aligned}&dui_A > df_A + dg_A; \end{aligned}$$

(10)

$$\begin{aligned}&deu_B > df_B + dg_B; \end{aligned}$$

(11)

$$\begin{aligned}&df_B >doi_B + dg_B. \end{aligned}$$

(12)

Proof

Using the notation form of the proof of the previous lemma, we assume that (i, j) is a pure-strategy Nash equilibrium precisely for each $i \in \{1,2\}$ and $j \in \{ 1,\ \ldots ,\ 8\}$, and we prove the conditions (9), (10), (11), and (12):

• Conditions of Lemma 1 are satisfied, therefore (9).

• The strategy (5, 11) is not a pure-strategy Nash equilibrium. Therefore, one of the following conditions should be satisfied:

  • (a) $\pi _A (i,11) > \pi _A (5,11)$ for $i\in \{1,\ \ldots ,\ 4\}$, expanded $f_A'+g_A'+u_A> f_A+g_A'+o_A$;

  • (b) $\pi _A (i,11) > \pi _A (5,11)$ for $i\in \{6,\ \ldots ,\ 8\}$, expanded $f_A+g_A'+o_A> f_A+g_A'+o_A$;

  • (c) $\pi _B (5,j) > \pi _B (5,11)$ for $j\in \{1,2,3,4,9,10,12\}$, expanded $f_B'+g_B+o_B> f_B'+g_B+o_B$;

  • (d) $\pi _B (5,j) > \pi _B (5,11)$ for $j\in \{5,6,7,8,13,14,15,16\}$, expanded $f_B+g_B'+i_B> f'_B+g_B+o_B$.

  Conditions (b) and (c) are false, and Condition (a) is in contradiction with the assumptions $df_A>0$ and $dou_A>0$. Therefore, (d) holds, implying (12).

• The strategy (6, 14) is not a pure-strategy Nash equilibrium. Therefore, one of the following conditions should be satisfied:

  • (a) $\pi _A (i,14) > \pi _A (6,14)$ for $i\in \{1,3\}$, expanded $f_A'+g_A'+u_A> f_A+g_A+i_A$;

  • (b) $\pi _A (i,14) > \pi _A (6,14)$ for $i\in \{2,4\}$, expanded $f_A'+g_A+u_A> f_A+g_A+i_A$;

  • (c) $\pi _A (i,14) > \pi _A (6,14)$ for $i\in \{5,7\}$, expanded $f_A+g_A'+i_A> f_A+g_A+i_A$;

  • (d) $\pi _A (i,14) > \pi _A (6,14)$ for $i\in \{8\}$, expanded $f_A+g_A+i_A> f_A+g_A+i_A$;

  • (e) $\pi _B (5,j) > \pi _B (6,14)$ for $j\in \{1,2,3,4,9,10,11,12\}$, expanded $f'_B+g_B+o_B> f_B+g_B+i_B$;

  • (f) $\pi _B (5,j) > \pi _B(6,14)$ for $j\in \{5,6,7,8,13,15,16\}$, expanded $f_B+g_B+i_B> f_B+g_B+i_B$.

  Conditions (d) and (f) are false, and Condition (c) is in contradiction with the assumption $dg_A>0$. Condition (e) is in contradiction with (12). Condition (b) implies (a), therefore (a) holds, implying (10).

• The strategy (4, 16) is not a pure-strategy Nash equilibrium. Therefore, one of the conditions:

  • (a) $\pi _A (i,16) > \pi _A (4,16)$ for $i\in \{1, 3\}$, expanded $f'_A+g_A'+u_A> f'_A+g_A+u_A$;

  • (b) $\pi _A (i,16) > \pi _A (4,16)$ for $i\in \{ 2\}$, expanded $f'_A+g_A+u_A> f'_A+g_A+u_A$;

  • (c) $\pi _A (i,16) > \pi _A (4,16)$ for $i\in \{5,\ \ldots ,\ 8\}$, expanded $f_A+g_A'+i_A> f'_A+g_A+u_A$;

  • (d) $\pi _B (4,j) > \pi _B (4,16)$ for $j\in \{1,\ \ldots ,\ 8\}$, expanded $f'_B+g_B'+e_B> f_B+g_B+u_B$;

  • (e) $\pi _B (4,j) > \pi _B (4,16)$ for $j\in \{9,\ \ldots ,\ 16\}$, expanded $f_B+g_B+u_B> f_B+g_B+u_B$; should be satisfied.

  Conditions (b) and (e) are false, and Condition (a) is in contradiction with the assumption $dg_A>0$. Condition (c) is in contraction with (10). Therefore, (d) must hold, implying (11).

For the converse implication, we assume the conditions (9), (10), (11), and (12), and we need to prove that (i, j), where $i\in \{1,2\}$ and $j\in \{1,\ \ldots ,\ 8\}$ are the only Nash equilibria. Therefore, for each $(m,n)\notin \{1,2\}\times \{1,\ \ldots ,\ 8\}$, we establish one of

$$\begin{aligned} \exists k\in \{1,\ \ldots ,\ 16\}: \pi _A(k,n)> \pi _A(m,n); \end{aligned}$$

(13)

or

$$\begin{aligned} \exists l \in \{1,\ \ldots ,\ 8\}: \pi _B(m,l)> \pi _B(m,n) . \end{aligned}$$

(14)

• 1. Condition (13) is satisfied for the pairs of strategies in the following cases:

  • (a) (10) implies $f'_A+g_A'+u_A>f_A+g_A+i_A$, therefore $\pi _A(k,l)> \pi _A(m,l)$ for $k\in \{1,3\}$, $l \in \{14\}$, and $m \in \{6,8\}$.

  • (b) $dg_A> 0$ implies $f'_A+g_A+e_A>f'_A+g_A'+e_A$, therefore $\pi _A(k,l)> \pi _A(m,l)$ for $k\in \{1,2\}$, $l \in \{1,\ \ldots ,\ 8\}$, and $m \in \{3,4\}$.

    • i. $dg_A>0$ with (9) implies $f'_A+g_A+e_A>f_A+g_A'+o_A$, therefore $\pi _A(k,l)> \pi _A(m,l)$ for $k\in \{1,2\}$ and either $l \in \{1,3,4\}$ and $m \in \{5,\ \ldots ,\ 8\}$; or $l \in \{2\}$ and $m \in \{7,8\}$.

    • ii. $dg_A> 0$ with (10) implies $f'_A+g_A'+u_A>f_A+g'_A+i_A$, therefore $\pi _A(k,l)> \pi _A(m,l)$ for $k\in \{2,4\}$ and either $l \in \{13,15,16\}$ and $m \in \{5,\ \ldots ,\ 8\}$; or $l \in \{14\}$ and $m \in \{5,7\}$.

  • (c) $dei_A> deo_A$ with (9) implies $f'_A+g_A+e_A>f_A+g_A+i_A$, therefore $\pi _A(k,l)> \pi _A(m,l)$ for $k\in \{1,2\}$, $l \in \{6\}$, and $m \in \{6,8\}$.

    • i. $dg_A>0$ and $dei_A> deo_A$ with (9) imply $f'_A+g_A+e_A>f_A+g'_A+i_A$, therefore $\pi _A(k,l)> \pi _A(m,l)$ for $k\in \{1,2\}$ and both $l \in \{6\}$ and $m \in \{5,7\}$, as well as $l \in \{5,7,8\}$ and $m \in \{5,\ \ldots ,\ 8\}$.

• 2. Condition (14) is satisfied for the strategies in the following cases:

  • (a) (12) implies $f_B+g'_B+i_B>f'_B+g_B+o_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k\in \{5\}$, $l \in \{5,\ \ldots ,\ 8\}$ and $n\in \{ 2,9,10,11,12\}$.

    • i. $dg_B>0$ with (12) implies $f_B+g'_B+i_B>f'_B+g'_B+o_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k\in \{7\}$, $l\in \{5,\ \ldots ,\ 8\}$, and $n\in \{ 9,\ \ldots ,\ 12\}$.

    • ii. $dg_B>0$ with (12) implies $f_B+g_B+i_B>f'_B+g'_B+o_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k\in \{8\}$, $l\in \{5,\ \ldots ,\ 8\}$, and $n\in \{ 9,\ \ldots ,\ 12\}$.

    • iii. $dg_B>0$ with (12) implies $f_B+g_B+i_B>f'_B+g_B+o_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k\in \{6\}$, $l\in \{5,\ \ldots ,\ 8\}$, and $n\in \{ 2, 9,10,11,12\}$.

  • (b) (11) and $dg_B>0$ imply $f'_B+g_B+e_B>f_B+g_B+u_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k \in \{2\}$, $l\in \{1,\ \ldots ,\ 8\}$, and $n\in \{9,\ \ldots ,\ 16\}$.

  • (c) (11) implies $f'_B+g_B'+e_B>f_B+g_B+u_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k \in \{ 4\}$, $l\in \{1,\ \ldots ,\ 8\}$, and $n\in \{9,\ \ldots ,\ 16\}$.

    • i. $dg_B>0$ with (11) implies $f'_B+g_B+e_B>f_B+g'_B+u_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k \in \{ 1\}$, $l\in \{1,\ \ldots ,\ 8\}$, and $n\in \{ 9,\ \ldots ,\ 16 \}$.

    • ii. $dg_B>0$ with (11) implies $f'_B+g'_B+e_B>f_B+g'_B+u_B$, therefore $\pi _B(k,l)>\pi _B(k,n)$ for $k \in \{ 3\}$, $l\in \{1,\ \ldots ,\ 8\}$, and $n\in \{ 9,\ \ldots ,\ 16 \}$.

Table 6 Arguments for a specific pair of strategies not being Nash equilibria

We have shown for each $(m,n)\notin \{1,2\}\times \{1,\ \ldots ,\ 8\}$ that the pair of strategies (m, n) is not a Nash equilibrium, and we summarize the arguments in Table 6.

Note that if all the pure-strategy Nash equilibria exhibit the desired outcome, so will all mixed-strategy Nash equilibria. Thus each Nash equilibrium of the Game yields the desired outcome, whenever the necessary and sufficient conditions from Theorem 1 are satisfied.