Non-Cooperative Bargaining with Unsophisticated Agents

Abstract

A traditional non-cooperative bargaining situation involves two or more forward-looking players making offers and counteroffers alternately until an agreement is reached, with a penalty according to the time taken by players in the decision-making process. We introduce a game that aids myopic players to reach the equilibrium as if they were forward-looking agents. The key elements of the game are that players are penalized both for their deviation from the previous best-reply strategy and their time taken for the decision-making at each step of the game. It is shown that our game has an equilibrium not only for the traditional processes and utilities used in traditional non-cooperative bargaining literature, but for an expanded and very comprehensive set of stochastic processes (such as Markov processes) and utility functions. Our work not only complements traditional non-cooperative bargaining literature for myopic agents, but also enlarges the class of processes and functions where Rubinstein’s non-cooperative bargaining solutions might be defined and applied.

Extensions

We now present two extensions of the bargaining game provided above that include the case when agents have different discount factors, and another where agents might coordinate on their demands. The convergence of results follows trivially from our general analysis presented above.

1.1 Bargaining Under Different Discounting

In this approach we present a solution where at each step of the negotiation process players calculate the Nash equilibrium considering the utility functions of all players but with the particularity that internally each player reaches this equilibrium point in a different time. Following the description of the model presented previously, we redefine the advantage of propose a new offer that depends on the utility function

$$\begin{aligned} f(x_{t},x_{t+1}):=\sum \limits _{\iota =1}^{{\mathfrak {n}}}\left[ \psi ^{\iota }(x_{t+1})-\psi ^{\iota }(x_{t})\right] \ge 0 \end{aligned}$$

for all players to reject the offer $x_t$ and making a new offer $x_{t+1}$ given the time spent to benefit of this advantage $T(x_{t+1})>0$, and $\alpha ^{\iota } (x_{t})$ be the weight that players put on their advantages to reject the offer $x_t$. Thus, the advantages to reject the offer $x_{t}$ and to propose a new offer $x_{t+1}$ are given by $A(x_{t},x_{t+1})=\alpha (x_{t})T(x_{t+1})f(x_{t},x_{t+1})$.

Remark 4

The function $f(x_{t},x_{t+1})$ satisfies the Nash condition

$$\begin{aligned} \begin{array}{c} \psi ^{\iota }(x_{t+1}) - \psi ^{\iota }(x_{t}) \ge 0 \end{array} \end{aligned}$$

for any $x\in X$ and all players.

Definition 7

A strategy $x^{*}\in X$ is said to be a Nash equilibrium if

$$\begin{aligned} x^{*}{\text { } \in }\text {Arg} \max _{x\in X}\text { }\left\{ f(x_{t},x_{t+1}) \right\} \end{aligned}$$

Then, at each step of the bargaining game we have in proximal format that the players must select their strategies according to

$$\begin{aligned} x^{* }= \arg \underset{x\in X}{\max }\left\{ - \delta _{t}T(x)\left\| \left( x-x^{* }\right) \right\| ^{2}+ \alpha _{t}T(x) f(x,x^*) \right\} \end{aligned}$$

(19)

where

$$\begin{aligned} f(x,x^*):=\sum \limits _{\iota =1}^{{\mathfrak {n}}}\left[ \psi ^{\iota }(x)-\psi ^{\iota }(x^*)\right] \end{aligned}$$

At each step of the bargaining process, players calculate simultaneously the Nash equilibrium but considering that each player reach the equilibrium in a different time.

1.1.1 Markov Chains

Let us to define the Nash equilibrium as a strategy $x^{* }=\left( x^{1* },\ldots ,x^{{\mathfrak {n}}}\right)$ such that

$$\begin{aligned} {\psi }\left( x^{1* },\ldots ,x^{{\mathfrak {n}}* }\right) \ge {\psi }\left( x^{1* },\ldots ,x^{\iota },\ldots ,x^{{\mathfrak {n}}* }\right) \end{aligned}$$

for any $x^{\iota }\in X$.

Consider that players try to reach the Nash equilibrium of the bargaining problem, that is, to find a joint strategy $x^{* }=\left( x^{1* },\ldots ,x^{{\mathfrak {n}}* }\right)$ $\in$ X satisfying for any admissible $x^{\iota }\in X^{\iota }$ and any $\iota =\overline{1,{\mathfrak {n}}}$

$$\begin{aligned} f( x,{\hat{x}}(x)) := \sum \limits _{\iota =1}^{{\mathfrak {n}}} \left[ \psi ^{\iota }\left( x^{\iota },x^{{\hat{l}}}\right) - \psi ^{\iota }\left( {\bar{x}}^{\iota },x^{{\hat{l}}}\right) \right] \end{aligned}$$

(20)

where ${\hat{x}}=(x^{{\hat{1}}\top },\ldots ,x^{\mathfrak {{\hat{n}}}\top })^{\top }\in {\hat{X}}\subseteq {\mathbb {R}}^{{\mathfrak {n}}({\mathfrak {n}}-1)}$, ${\bar{x}}^{\iota }$ is the utopia point defined as Eq. (11) and $\psi ^{\iota }\left( x^{\iota },x^{{\hat{\iota }}}\right)$ is the concave cost-function of player $\iota$ which plays the strategy $x^{\iota }\in X^{\iota }$ and the rest of players the strategy $x^{{\hat{\iota }}}\in X^{{\hat{\iota }}}$ defined as Eq. (16) considering the time function.

Remark 5

The function $f( x,{\hat{x}}(x))$ satisfies the Nash condition

$$\begin{aligned} \begin{array}{c} \psi ^{\iota }\left( x^{\iota },x^{{\hat{\iota }}}\right) - \psi ^{\iota }\left( {\bar{x}}^{\iota },x^{{\hat{\iota }}}\right) \le 0 \end{array} \end{aligned}$$

(21)

for any $x^{\iota }\in X^{\iota }$ and all $\iota =\overline{1,{\mathfrak {n}}}$

Definition 8

A strategy $x^{*}\in X$ is said to be a Nash equilibrium if

$$\begin{aligned} { x}^{*}{\text { } \in } \text { Arg} \max _{x\in X_{\text {adm}}}\text { }\left\{ f( x,{\hat{x}}(x)) \right\} \end{aligned}$$

Remark 6

If $f( x,{\hat{x}}(x))$ is strictly concave then

$$\begin{aligned} { x}^{*}{\text { } = \text { }}\arg \max _{x\in X_{\text {adm}}}\text { }\left\{ f( x,{\hat{x}}(x)) \right\} \end{aligned}$$

We redefine the utility function that depends of the average utility function of all players as follows

$$\begin{aligned} \begin{array}{c} F(x,{\hat{x}}(x)):= f( x,{\hat{x}}(x)) - \frac{1}{2} \sum \limits _{\iota =1}^{{\mathfrak {n}}}\sum \limits _{j=1}^{N}\mu _{(j)}^{\iota }h_{(j)}^{\iota }(x^{\iota })- \\ \\ \frac{1}{2} \sum \limits _{\iota =1}^{{\mathfrak {n}}}\sum \limits _{i=1}^{N}\sum \limits _{j=1}^{N} \sum \limits _{k=1}^{M}\xi _{(j)}^{\iota }q_{(j|i,k)}^{\iota }{x}_{(i,k)}^{\iota } - \frac{1}{2} \sum \limits _{\iota =1}^{{\mathfrak {n}}}\sum \limits _{i=1}^{N}\sum \limits _{k=1}^{M} \eta ^{\iota }\left( {x}_{(i,k)}^{\iota }-1\right) \end{array} \end{aligned}$$

then, we may conclude that

$$\begin{aligned} x^*=\arg \underset{x\in X,{\hat{x}}\in {\hat{X}}}{\max }\quad \underset{\mu \ge 0,\xi \ge 0,\eta \ge 0}{\min }\quad F(x,{\hat{x}}(x),\mu ,\xi ,\eta ) \end{aligned}$$

(22)

Finally we have that at each step of the bargaining process, players calculate the Nash equilibrium (but they reach the equilibrium at different time) according to the solution of the non-cooperative bargaining problem in proximal format defined as follows

$$\begin{aligned} \begin{array}{c} \mu ^{* }=\arg \underset{\mu \ge 0}{\min }\left\{ - \delta \Vert \mu -\mu ^{* }\Vert ^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),\mu ,\xi ^*,\eta ^*\right) \right\} \\ \xi ^{* }=\arg \underset{\xi \ge 0}{\min }\left\{ - \delta \Vert \xi -\xi ^{* }\Vert ^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),\mu ^*,\xi ,\eta ^*\right) \right\} \\ \eta ^{* }=\arg \underset{\eta \ge 0}{\min }\left\{ - \delta \Vert \eta -\eta ^{* }\Vert ^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),\mu ^*,\xi ^*,\eta \right) \right\} \\ x^{* }=\arg \underset{x\in X}{\max }\left\{ - \delta \left\| \left( x-x^{* }\right) \right\| _{\varLambda }^{2}+\alpha F\left( x,{\hat{x}}^*(x),\mu ^*,\xi ^*,\eta ^*\right) \right\} \\ {\hat{x}}^{* }=\arg \underset{{\hat{x}}\in {\hat{X}}}{\max }\left\{ - \delta \left\| \left( {\hat{x}}-{\hat{x}}^{* }\right) \right\| _{\varLambda }^{2}+\alpha F\left( x^*,{\hat{x}}(x),\mu ^*,\xi ^*,\eta ^*\right) \right\} \end{array} \end{aligned}$$

(23)

1.1.2 Transfer Pricing Simulation

Following the Section above, in this model each player calculates the strategies according the Nash equilibrium formulation where players calculate the Nash equilibrium simultaneously, but with the characteristic that they reach the equilibrium at different time, following the relation (23) until they reach an agreement (strategies show convergence). Figures 11, 12 and 13 show the behavior of the offers (strategies) during the bargaining process.

Fig. 11

Strategies of player 1 in the bargaining model 2

Fig. 12

Strategies of player 2 in the bargaining model 2

Fig. 13

Strategies of player 3 in the bargaining model 2

Fig. 14

Behavior of players’ utilities in the bargaining model 2

Finally, the agreement reached is as follows:

$$\begin{aligned} \begin{array}{ccccc} c^{1 }= \begin{bmatrix} 0.2127 &{} 0.0074 \\ 0.1429 &{} 0.0050 \\ 0.6106 &{} 0.0214 \end{bmatrix} &{} \quad &{} c^{2 }= \begin{bmatrix} 0.0050 &{} 0.2366 \\ 0.0087 &{} 0.4117 \\ 0.0070 &{} 0.3310 \end{bmatrix} &{} \quad &{} c^{3 }= \begin{bmatrix} 0.2237 &{} 0.0071 \\ 0.5877 &{} 0.0186 \\ 0.1579 &{} 0.0050 \end{bmatrix} \end{array} \end{aligned}$$

Following (6) the mixed strategies obtained for players are as follows

$$\begin{aligned} \begin{array}{ccccc} d^{1 }= \begin{bmatrix} 0.9662 &{} 0.0338 \\ 0.9662 &{} 0.0338 \\ 0.9662 &{} 0.0338 \end{bmatrix} &{} \quad &{} d^{2 }= \begin{bmatrix} 0.0207 &{} 0.9793 \\ 0.0207 &{} 0.9793 \\ 0.0207 &{} 0.9793 \end{bmatrix} &{} \quad &{} d^{3 }= \begin{bmatrix} 0.9693 &{} 0.0307 \\ 0.9693 &{} 0.0307 \\ 0.9693 &{} 0.0307 \end{bmatrix} \end{array} \end{aligned}$$

With the strategies calculatedat each step of the negotiation process, the utilities of each player showed a decreasing behavior as shown in the Fig. 14, i.e., at each step of the bargaining process, the utility of each player decreases until they reach an agreement. At the end of the bargaining process, the resulting utilities are as follows $\psi ^{1}(c^1,c^2,c^3)=986.8936$, $\psi ^{2}(c^1,c^2,c^3)=651.4633$ and $\psi ^{2}(c^1,c^2,c^3)=949.6980$ for each player.

1.2 Bargaining with Collusive Behavior

In this approach we analyze a bargaining situation where players make groups and alternately each group makes an offer to the others until they reach an equilibrium point (agreement). We describe a bargaining model with two teams of players as follows. Let us consider a bargaining game with ${\mathfrak {n}}+{\mathfrak {m}}$ players. Let ${\mathcal {N}}=\{1,\ldots ,{\mathfrak {n}}\}$ denote the set of players called team A and let us define the behavior of all players $\iota =\overline{1,{\mathfrak {n}}}$ as $x_t=(x_t^1,\ldots ,x_t^{\mathfrak {n}}) \in X$ where X is a convex and compact set. In the same way, the rest ${\mathcal {M}}=\{1,\ldots ,{\mathfrak {m}}\}$ players are the team B and let the set of the strategy profiles of all player $m=\overline{1,{\mathfrak {m}}}$ be defined by $y_t=(y_t^1,\ldots ,y_t^{\mathfrak {m}}) \in Y$ where Y is a convex and compact set. Then, $X \times Y$ in the set of full strategy profiles. In this model the function $\psi (x,y)$ represents the utility function of team A which determines the decision of accept or reject the offer; similarly, team B makes the decision according to its utility function $\varphi (x,y)$.

Following the description of the model presented above, we redefine the advantage of propose a new offer considering the utility function for team A as follows

$$\begin{aligned} f(x_{t},y_{t},x_{t+1},y_{t+1}):=\sum \limits _{\iota =1}^{{\mathfrak {n}}}\left[ \psi ^{\iota }(x_{t+1},y_{t})-\psi ^{\iota }(x_{t},y_{t})\right] \ge 0 \end{aligned}$$

and, similarly the utility function for team B is as follows

$$\begin{aligned} g(x_{t},y_{t},x_{t+1},y_{t+1}):=\sum \limits _{m=1}^{{\mathfrak {m}}}\left[ \varphi ^{\iota }(x_{t},y_{t+1})-\varphi ^{\iota }(x_{t},y_{t})\right] \ge 0 \end{aligned}$$

Thus, the advantages for team A to reject the offer $x_{t}$ and to propose a new offer $x_{t+1}$ are given by $A(x_{t},y_{t},x_{t+1},y_{t+1})=\alpha (x_{t})T(x_{t+1})f(x_{t},y_{t},x_{t+1},y_{t+1})$; in the same way, the advantages for team B to reject the offer $y_{t}$ and to propose a new offer $y_{t+1}$ are given by $A(x_{t},y_{t},x_{t+1},y_{t+1})=\alpha (y_{t})T(y_{t+1})g(x_{t},y_{t},x_{t+1},y_{t+1})$.

Remark 7

The function $f(x_{t},y_{t},x_{t+1},y_{t+1})$ satisfies the Nash condition

$$\begin{aligned} \begin{array}{c} \psi ^{\iota }(x_{t+1},y_{t})-\psi ^{\iota }(x_{t},y_{t}) \ge 0 \end{array} \end{aligned}$$

for any $x\in X$, $y\in Y$ and $\iota =\overline{1,{\mathfrak {n}}}$ players.

Remark 8

The function $g(x_{t},y_{t},x_{t+1},y_{t+1})$ satisfies the Nash condition

$$\begin{aligned} \begin{array}{c} \varphi ^{\iota }(x_{t},y_{t+1})-\varphi ^{\iota }(x_{t},y_{t}) \ge 0 \end{array} \end{aligned}$$

for any $x\in X$, $y\in Y$ and $m=\overline{1,{\mathfrak {m}}}$ players.

The dynamics of the bargaining game is as follows: at each step of the negotiation process the team A chooses a strategy $x \in X$ considering the utility function $f(x_{t},y_{t},x_{t+1},y_{t+1})$, then team B must decide between to accept or reject the offer calculating a new offer (strategies) $y \in Y$ considering the utility function of the group $g(x_{t},y_{t},x_{t+1},y_{t+1})$. Following the description of the model 1, now we have that teams solve the problem in proximal format as follows:

$$\begin{aligned} \begin{array}{c} x^{* }= \arg \underset{x\in X}{\max }\left\{ - \delta _{t}T(x)\left\| \left( x-x^{* }\right) \right\| ^{2}+ \alpha _{t}T(x) f(x,y,x^*,y^*) \right\} \\ \\ y^{* }= \arg \underset{y\in Y}{\max }\left\{ - \delta _{t}T(y)\left\| \left( y-y^{* }\right) \right\| ^{2}+ \alpha _{t}T(y) g(x,y,x^*,y^*) \right\} \end{array} \end{aligned}$$

(24)

where

$$\begin{aligned} \begin{array}{c} f(x,y,x^*,y^*):=\sum \limits _{\iota =1}^{{\mathfrak {n}}}\left[ \psi ^{\iota }(x,y^*)-\psi ^{\iota }(x^*,y^*)\right] \\ \\ g(x,y,x^*,y^*):=\sum \limits _{m=1}^{{\mathfrak {m}}}\left[ \varphi ^m(x^*,y)-\varphi ^m(x^*,y^*)\right] \end{array} \end{aligned}$$

At each step, teams make a new offer according to Eq. (24), both teams solve the bargaining problem together but they reach the equilibrium at different time, the bargaining game continues until the offers (strategies) of all player show convergence.

1.2.1 Markov Chains

For this model, in the same way that we define the strategies $x \in X$, let us consider a set of strategies denoted by $y^{m}\in Y^{m}$ $\left( m=\overline{1,{\mathfrak {m}}}\right)$ where $Y:=\bigotimes \limits _{m=1}^{{\mathfrak {m}}}Y^{\iota }$ is a convex and compact set,

$$\begin{aligned} y^{m}:=\text {col }(c^{m}),\quad Y^{m}:=C_{\text {adm}}^{m} \end{aligned}$$

where col is the column operator.

Denote by $y=(y^{1},\ldots ,y^{{\mathfrak {m}}})^{\top }\in Y$, the joint strategy of the players and $y^{{\hat{m}}}$ is a strategy of the rest of the players adjoint to $y^{m}$, namely,

$$\begin{aligned} y^{{\hat{m}}}:=\left( y^{1},\ldots ,y^{m-1},y^{m+1},\ldots ,y^{{\mathfrak {m}}}\right) ^{\top }\in Y^{{\hat{m}}}:=\bigotimes \limits _{h=1,\text { }h\ne m}^{{\mathfrak {m}}}Y^{h} \end{aligned}$$

such that $y=(y^{m},y^{{\hat{m}}})$, $m=\overline{1,{\mathfrak {m}}}$.

Consider that players of team A try to reach the Nash equilibrium of the bargaining problem, that is, to find a joint strategy $x^{* }=\left( x^{1* },\ldots ,x^{{\mathfrak {n}}* }\right)$ $\in$ X satisfying for any admissible $x^{\iota }\in X^{\iota }$ and any $\iota =\overline{1,{\mathfrak {n}}}$

$$\begin{aligned} f( x,{\hat{x}}(x)|y) := \sum \limits _{\iota =1}^{{\mathfrak {n}}} \left[ \psi ^{\iota }\left( x^{\iota },x^{{\hat{l}}}|y\right) - \psi ^{\iota }\left( {\bar{x}}^{\iota },x^{{\hat{\iota }}}|y\right) \right] \end{aligned}$$

(25)

where ${\hat{x}}=(x^{{\hat{1}}\top },\ldots ,x^{\mathfrak {{\hat{n}}}\top })^{\top }\in {\hat{X}}\subseteq {\mathbb {R}}^{{\mathfrak {n}}({\mathfrak {n}}-1)}$, ${\bar{x}}^{\iota }$ is the utopia point defined as Eq. (11) and $\psi ^{\iota }\left( x^{\iota },x^{{\hat{\iota }}}|y\right)$ is the concave cost-function of player $\iota$ which plays the strategy $x^{\iota }\in X^{\iota }$ and the rest of players the strategy $x^{{\hat{\iota }}}\in X^{{\hat{\iota }}}$ fixing the strategies $y \in Y$ of team B, and it is defined as Eq. (16) considering the time function.

Similarly, consider that players of team B also try to reach the Nash equilibrium of the bargaining problem, that is, to find a joint strategy $y^{* }=\left( y^{1* },\ldots ,y^{{\mathfrak {m}}* }\right)$ $\in$ Y satisfying for any admissible $y^{m}\in Y^{m}$ and any $m=\overline{1,{\mathfrak {m}}}$

$$\begin{aligned} g( y,{\hat{y}}(y)|x) := \sum \limits _{m=1}^{{\mathfrak {m}}} \left[ \psi ^{m}\left( y^{m},y^{{\hat{m}}}|x\right) - \psi ^{m}\left( {\bar{y}}^{m},y^{{\hat{m}}}|x\right) \right] \end{aligned}$$

(26)

where ${\hat{y}}=(y^{{\hat{1}}\top },\ldots ,y^{\mathfrak {{\hat{m}}}\top })^{\top }\in {\hat{Y}}\subseteq {\mathbb {R}}^{{\mathfrak {m}}({\mathfrak {m}}-1)}$, ${\bar{y}}^{m}$ is the utopia point defined as Eq. (11) and $\psi ^{m}\left( y^{m},y^{{\hat{m}}}|x\right)$ is the concave cost-function of player m which plays the strategy $y^{m}\in Y^{m}$ and the rest of players the strategy $y^{{\hat{m}}}\in Y^{{\hat{m}}}$ fixing the strategies $x \in X$ of team A, and it is defined as Eq. (16) considering the time function.

Then, we have that a strategy $x^* \in X$ of team A together with the collection $y^* \in Y$ of team B are defined as the equilibrium of a strictly concave bargaining problem if

$$\begin{aligned} ({x}^{*},{y}^{*}){\text { } = \text { }}\arg \max _{x\in X_{\text {adm}},y\in Y_{adm}}\text { }\left\{ f(x,{\hat{x}}(x)|y) \le 0 , g(y,{\hat{y}}(y)|x) \le 0 \right\} \end{aligned}$$

We redefine the utility function that depends of the average utility function of all players as follows

$$\begin{aligned} \begin{array}{c} F(x,{\hat{x}}(x),y,{\hat{y}}(y)):= f(x,{\hat{x}}(x)|y) + g(y,{\hat{y}}(y)|x) - \frac{1}{2} \sum \limits _{\iota =1}^{{\mathfrak {n}}}\sum \limits _{j=1}^{N}\mu _{(j)}^{\iota }h_{(j)}^{\iota }(x^{\iota })- \\ \\ \frac{1}{2} \sum \limits _{m=1}^{{\mathfrak {m}}}\sum \limits _{j=1}^{N}\mu _{(j)}^{m}h_{(j)}^{m}(y^{m})- \frac{1}{2} \sum \limits _{\iota =1}^{{\mathfrak {n}}}\sum \limits _{i=1}^{N}\sum \limits _{j=1}^{N} \sum \limits _{k=1}^{M}\xi _{(j)}^{\iota }q_{(j|i,k)}^{\iota }{x}_{(i,k)}^{\iota } - \frac{1}{2} \sum \limits _{m=1}^{{\mathfrak {m}}}\sum \limits _{i=1}^{N}\sum \limits _{j=1}^{N} \sum \limits _{k=1}^{M}\xi _{(j)}^{m}q_{(j|i,k)}^{m}{y}_{(i,k)}^{m} - \\ \\ \frac{1}{2} \sum \limits _{\iota =1}^{{\mathfrak {n}}}\sum \limits _{i=1}^{N}\sum \limits _{k=1}^{M} \eta ^{\iota }\left( {x}_{(i,k)}^{\iota }-1\right) - \frac{1}{2} \sum \limits _{m=1}^{{\mathfrak {m}}}\sum \limits _{i=1}^{N}\sum \limits _{k=1}^{M} \eta ^{m}\left( {y}_{(i,k)}^{m}-1\right) \end{array} \end{aligned}$$

then, we may conclude that

$$\begin{aligned} (x^*,y^*)=\arg \underset{x\in X,{\hat{x}}\in {\hat{X}},y\in Y,{\hat{y}}\in {\hat{Y}}}{\max }\quad \underset{\mu \ge 0,\xi \ge 0,\eta \ge 0}{\min }\quad F(x,{\hat{x}}(x),y,{\hat{y}}(y),\mu ,\xi ,\eta ) \end{aligned}$$

(27)

Finally we have that at each step of the bargaining process, players calculate their equilibrium according to the solution of the non-cooperative bargaining problem in proximal format defined as follows

$$\begin{aligned} \begin{array}{c} \mu ^{* }=\arg \underset{\mu \ge 0}{\min }\left\{ - \delta \Vert \mu -\mu ^{* }\Vert ^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),y^*,{\hat{y}}^*(y),\mu ,\xi ^*,\eta ^*\right) \right\} \\ \xi ^{* }=\arg \underset{\xi \ge 0}{\min }\left\{ - \delta \Vert \xi -\xi ^{* }\Vert ^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),y^*,{\hat{y}}^*(y),\mu ^*,\xi ,\eta ^*\right) \right\} \\ \eta ^{* }=\arg \underset{\eta \ge 0}{\min }\left\{ - \delta \Vert \eta -\eta ^{* }\Vert ^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),y^*,{\hat{y}}^*(y),\mu ^*,\xi ^*,\eta \right) \right\} \\ x^{* }=\arg \underset{x\in X}{\max }\left\{ - \delta \left\| \left( x-x^{* }\right) \right\| _{\varLambda }^{2}+\alpha F\left( x,{\hat{x}}^*(x),y^*,{\hat{y}}^*(y),\mu ^*,\xi ^*,\eta ^*\right) \right\} \\ {\hat{x}}^{* }=\arg \underset{{\hat{x}}\in {\hat{X}}}{\max }\left\{ - \delta \left\| \left( {\hat{x}}-{\hat{x}}^{* }\right) \right\| _{\varLambda }^{2}+\alpha F\left( x^*,{\hat{x}}(x),y^*,{\hat{y}}^*(y),\mu ^*,\xi ^*,\eta ^*\right) \right\} \\ y^{* }=\arg \underset{y\in Y}{\max }\left\{ - \delta \left\| \left( y-y^{* }\right) \right\| _{\varLambda }^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),y,{\hat{y}}^*(y),\mu ^*,\xi ^*,\eta ^*\right) \right\} \\ {\hat{y}}^{* }=\arg \underset{{\hat{y}}\in {\hat{Y}}}{\max }\left\{ - \delta \left\| \left( {\hat{y}}-{\hat{y}}^{* }\right) \right\| _{\varLambda }^{2}+\alpha F\left( x^*,{\hat{x}}^*(x),y^*,{\hat{y}}(y),\mu ^*,\xi ^*,\eta ^*\right) \right\} \end{array} \end{aligned}$$

(28)

1.2.2 Transfer Pricing Simulation

For this example, the team 1 is only formed by player 1 while team 2 is composed of players 2 and 3. Although the players calculate the strategies together following the relation (28), we consider that players reach the equilibrium at different times. Figures 15, 16 and 17 show the behavior of the offers (strategies) during the bargaining process.

Fig. 15

Strategies of player 1 in the bargaining model 3

Fig. 16

Strategies of player 2 in the bargaining model 3

Fig. 17

Strategies of player 3 in the bargaining model 3

Fig. 18

Behavior of players’ utilities in the bargaining model 3

Finally, the agreement reached is as follows:

$$\begin{aligned} \begin{array}{ccccc} c^{1 }= \begin{bmatrix} 0.2127 &{} 0.0074 \\ 0.1429 &{} 0.0050 \\ 0.6106 &{} 0.0214 \end{bmatrix} &{} \quad &{} c^{2 }= \begin{bmatrix} 0.0050 &{} 0.2366 \\ 0.0087 &{} 0.4117 \\ 0.0070 &{} 0.3310 \end{bmatrix} &{} \quad &{} c^{3 }= \begin{bmatrix} 0.2237 &{} 0.0071 \\ 0.5877 &{} 0.0186 \\ 0.1579 &{} 0.0050 \end{bmatrix} \end{array} \end{aligned}$$

Following (6) the mixed strategies obtained for players are as follows

$$\begin{aligned} \begin{array}{ccccc} d^{1 }= \begin{bmatrix} 0.9662 &{} 0.0338 \\ 0.9662 &{} 0.0338 \\ 0.9662 &{} 0.0338 \end{bmatrix} &{} \quad &{} d^{2 }= \begin{bmatrix} 0.0207 &{} 0.9793 \\ 0.0207 &{} 0.9793 \\ 0.0207 &{} 0.9793 \end{bmatrix} &{} \quad &{} d^{3 }= \begin{bmatrix} 0.9693 &{} 0.0307 \\ 0.9693 &{} 0.0307 \\ 0.9693 &{} 0.0307 \end{bmatrix} \end{array} \end{aligned}$$

With the strategies calculated at each step of the negotiation process, the utilities of each player showed a decreasing behavior as shown in the Fig. 18, i.e., at each step of the bargaining process, the utility of each player decreases until they reach an agreement. At the end of the bargaining process, the resulting utilities are as follows $\psi ^{1}(c^1,c^2,c^3)=986.8936$, $\psi ^{2}(c^1,c^2,c^3)=651.4631$ and $\psi ^{2}(c^1,c^2,c^3)=949.6978$ for each player.

The following figure shows the behavior of the utilities at each of the applied models (model 1 is the general bargaining model, model 2 corresponds to bargaining under different discounting and model 3 to bargaining with collusive behavior), we can see that the utilities begin at the same point, the strong Nash equilibrium, and then decrease until the strategies converge (see Fig. 19). From the results obtained we observed that model 1 favors the utilities of players 2 and 3, while model 2 and 3 are better for player 1. We also observed that even if models 2 and 3 reach the same agreement (equilibrium point) the strategies and, as a consequence, the utilities have a different behavior during the bargaining process.

Fig. 19

Behavior of the utilities at each model