Finding the Source of Negotiation Power: A Two-round Bargaining Model with Two-side Uncertainty

In this paper, we successfully established a solvable bargaining model and shows the source of negotiation power. We build a two-round bargaining model with both-side uncertainty to study the negotiation power. We construct a Sequential Equilibrium for this game and characterize it by an ordinary differential equation. Due to the computational difficulty of the SE solution, we further find a feasible but almost the same good solution, Optimal Commitment. Under this solution, we restate the strategy for both buyer and seller and analyze the negotiation power from the model's perspective. From our model, we obverse the following three factors that influence the negotiation power, proposing order, patience, and the bound of the opponent's base price.

Introduction

Negotiation is everywhere in our daily life. People buy things from the market. Students unions strive for sponsorship when holding activities. Companies negotiating on trades. Thus, having a deep understanding of negotiation can help us benefit more in negotiation activities in both life and work. I used to have a question that why the two sides do not directly make a deal on the middle point of their base price? More generally, where does the negotiation power come from?

Game theory researchers have studied negotiation and bargaining for many decades. However, the models that they studied are limited, for example, deterministic game, one-side uncertainty game, infinite-horizon game, continuous-time game, etc. [1, 2, 3] These models all have some limitations. So my target is extracting the most important features about negotiating and establishing a solvable model.

Model

The model in this paper is a two-player Bayesian extensive-form game. A buyer and a seller are bargaining over the price of an object. The object is worth \(s\) to the seller and \(b\) to the buyer. The bargaining continues for \(T\) rounds, and the buyer and seller give a proposal in turn. In the \((2k-1)\)-th round \((k = 1, 2, \ldots)\), the seller propose a value \(p_{2k-1}\), and the buyer decide whether it agrees on this price. And in the \((2k)\)-th round \((k = 1, 2, \ldots)\), the buyer propose a value \(p_{2k}\), and the seller decide whether it agrees on this price. If someone agrees at some proposal, the game ends. Otherwise, the negotiation breaks if no consensus have made by the \(T\)-th round, and both buyer and seller profit \(0\).

To model the incentive of agreeing, we have two variables \(\delta_s, \delta_b\) which is the decay rate of profit for the seller and buyer. Specifically, suppose the buyer and the seller agree on price \(v\) on the \(i\)-th round, we have \(u_s(v, i) = \delta_s^{i-1} (v - s)\) and \(u_b(v, i) = \delta_b^{i-1} (b - v)\). The decay rates contain all factors related to time, for example, the opportunity cost for negotiating a long time, the impatience of the participants, etc.

In our model, we deem that the reserve price \(s\) and \(b\) obey some know distribution \(U(\underline{s}, \bar{s})\) and \(U(\underline{b}, \bar{b})\). Parameters \(\underline{s}, \bar{s}, \underline{b}, \bar{b}\) are common knowledge for both buyer and seller.

In this paper, we analyzed a simple case where \(T = 2\). The seller first proposes the price \(p_1\), and the buyer decides whether they deal on this price \(p_1\). If they agree on \(p_1\), the seller profit \(p_1 - s\) and the buyer profit \(b - p_1\). Otherwise, the buyer proposes the price \(p_2\), and the seller decides whether they deal on this price \(p_2\). If they agree on \(p_2\), the seller profit \(\delta_s(p_2 - s)\) and the seller profit \(\delta_b(b - p_2)\). If the two proposals are rejected, both of then profit 0.

Also, we assume \(\underline{s} \leq \underline{b} \leq \bar{s} \leq \bar{b}\) because this is the most common situation. The reserve price of the seller is generally lower than the expected price of the buyer, while sometimes the reserve price of the seller can higher than the expected price of the buyer. The other cases can use the same method to analyze and obtain an analogous result.

Analysis

Our target is to find a feasible strategy profile for the seller and buyer. Based on the strategy profile, we will confirm the power of negotiation and useful tricks in bargaining. First of all, we want to find a Sequential Equilibrium (SE). Sequential Equilibrium is just the Nash Equilibrium in Bayesian extensive form game, such that after every history each player's strategy is optimal given the other's strategy and his current belief. [1]

Using the Backward Induction idea and the increasing difference theorem, the framework of strategy is as follows. The seller proposed a price \(p_1 = x(s)\) according to its reserved price. The buyer decide whether it rejects or accepts by a cutoff strategy where it rejects if \(b < b^*(p_1)\) and accepts if \(b \geq b^*(p_1)\). Suppose the first proposal is rejected, the buyer proposed a price \(p_2 = y(b, x^{-1}(p_1))\) according to its expected value and the estimated seller's reserved value based on \(p_1\). Finally, the buyer decides whether it rejects or accepts by a cutoff strategy where it rejects if \(s < s^*(p_2)\) and accepts if \(s \geq s^*(p_2)\).

we first calculate the optimal \(y\) and \(s^*\). For the second proposal, given the proposed price, if the seller can profit a positive value, the seller should accept, and thus, \(s^* = y\). We assume the seller's proposal is an injective function of its reserved price. The buyer can estimate the seller's reserved price, denoted as \(\hat{s} = x^{-1}(p_1)\), by applying the inversed function. Given both \(\hat{s} = s\) and \(b\), the buyer should propose \(p_2 = y(b, \hat{s}) = \hat{s}\) in the second round.

Then we determine the cutoff strategy parameter \(b^*\) in the first round. Recall that the utility of the two options are equal when \(b = b^*\). So we have the equation, \[b^* - p_1 = \delta_b (b^* - p_2)\] The solution is that \(b^*(p_1) = \frac{p_1 - \delta_b \cdot \hat{s}}{1 - \delta_b}\).

Sequential Equilibrium

Until now, we have determined most of our strategy. However, finding the function \(x(s)\) of the price in the first round is the trickiest part.

Let \(\sigma\) denote the strategy profile that we construct. Note that \(\sigma\) including all parameters, \(x(s), y(b, \hat{s}), b^*, s^*\). Let \(u_s(s, b; \sigma)\) denote the utility of the seller when both player act according to our strategy \(\sigma\). We have \[ u_s(s, b; \sigma) = \left\{ \begin{array}{ll} 0, &b < b^* \\ x(s) - s, &b \geq b^* \end{array} \right. \] Furthermore, let \(v_s(s', s, b; \sigma)\) denote that utility of the seller when the seller diviate in the first round by proposing \(x(s')\), and the buyer still follows \(\sigma\). We have \[ v_s(s', s, b; \sigma) = \left\{ \begin{array}{ll} 0, &b < s' \\ \delta_s(s' - s), &s' \leq b < b^*(s') \\ x(s') - s, & b \geq b^*(s') \end{array} \right. \] Remark that suppose \(x \geq s\), then \(b^*(s) = \frac{x(s) - \delta_s \cdot s}{1 - \delta_s} \geq s\) (i.e. the step function \(v_s\) never degenerate). We omit \(\sigma\) in notation \(u_s\) and \(v_s\) in the following section since we never change it. We let \(u_s(s)\) and \(v_s(s', s)\) be the average of \(u_s(s, b)\) and \(v_s(s', s, b)\) over \(U(\underline{b}, \bar{b})\). \[ \begin{aligned} u_{s}(s) &= \frac{1}{\bar{b} - \underline{b}} \int_{\underline{b}}^{\bar{b}} u_{s}(s, b) db \\ v_{s}(s', s) &= \frac{1}{\bar{b} - \underline{b}} \int_{\underline{b}}^{\bar{b}} v_{s}(s', s, b) db \end{aligned} \]

Now, we consider the equilibrium restriction. Since the buyer would not deviate from \(\sigma\), the following equation holds. \[ v_{s}(s', s) < v_{s}(s, s), \qquad \forall s', s \in [\underline{s}, \bar{s}] \] By calculation, we obverse that \(v_{s}(s', s)\) is a quadratic function with respect to \(s'\). Thus, the equilibrium restriction is equivalent to \[ \left.\frac{dv_s}{ds'}(s', s)\right|_{s'=s} = 0 . \]

We now analyze the case that \(\underline{b} \leq s \leq \bar{b}\), and the other cases are simpler than this case and lead to the same result so we omit them in this paper. Consider deviating from \(s\) by \(ds\). Then the benefit of deviation is \[ \begin{aligned} v(s + ds, s) - u(s) &= \int_{\underline{b}}^{\bar{b}} v_{s}(s', s, b) db - \int_{\underline{b}}^{\bar{b}} u_{s}(s, b) db \\ &= (\hat{b} - b^*(s)) \cdot dx + \delta_s (b^*(s) - s) \cdot ds - (x - s) \frac{dx - \delta_b \cdot ds}{1 - \delta_b} \\ &= 0 \end{aligned} \] We rearrange the equation and get an ordinal differential equation. \[ x'(s) = \frac{(\delta_s + \delta_b)(x - s)}{2x - (1 - \delta_b) \cdot \bar{b} - (1 + \delta_b) \cdot s} \]

Unfortunately, we cannot find a simple open form for \(x(s)\). So a numerical case study is illustrated, where \(\underline{b} = \underline{s} = 0, \bar{s} = \bar{b} = 1, \delta_s = \delta_b = 0.5\). Then the differential equation turns to be \[ x'(s) = \frac{2x - 2s}{4x -3s - 1} \] Let \(x(1) = 1\) and plot \(x(s)\) by the differential equation. The curve is shown in Fig. 2. The curve looks like a straight line, nevertheless, it is a concave curve. Moreover, this differential equation is sensitive and has poor accuracy. These phenomena will be stated better in the next subsection.

Although we have characterized the equilibrium for this simple two-round bargaining, there are still some limitations. Firstly, our solution is not feasible because it is hard to calculate. Secondly, we cannot analyze the property deeper because we do not have a closed form. Therefore, we choose to study another feasible solution concept, Optimal Commitment.

Optimal Commitment: a feasible solution

Equilibrium is a very strong restriction. It requires that both players will not deviate if the strategy profile is common knowledge. However, in practice, people will not make their strategy public, and they focus on maximum profit rather than a stable point. Thus, Optimal Commitment is also a reasonable solution concept here, and we will see that it is more feasible.

In the Optimal Commitment solution, the first player commits its strategy and makes it public, and the second player best responds to the first player. The first player needs to find a strategy where it can profit most. Note that we only consider the optimal strategy in our strategy framework (i.e. \(p_1\) is an injection function with respect to \(s\)). Although we do not prove that the Optimal Commitment is an optimal strategy among all possible strategies, strategies in our strategy framework are all intuitively reasonable strategies.

Recall that \(u_s(s, b; \sigma)\) denote the utility of the seller when both players act according to our strategy \(\sigma\). Here, we consider value \(x(s)\) as a variable, \[ u_s(s, b; x(s), \sigma) = \left\{ \begin{array}{ll} 0, &b < b^* \\ x(s) - s, &b \geq b^* \end{array}\right. \] Therefore, we find the optimal initial price for each \(s\). \[ \begin{aligned} x(s) &= \mathop{argmax}_{x} \frac{1}{\bar{b} - \underline{b}}\int_{\underline{b}}^{\bar{b}} u_{s}(s, b; x) db \\ &= \mathop{argmax}_{x} \frac{1}{\bar{b} - \underline{b}} \int_{b^*(s)}^{\bar{b}} (x - s) db \\ &= \mathop{argmax}_{x} (x - s)(\bar{b} - b^*(s)) \\ &= \mathop{argmax}_{x} (x - s)(x - \delta_b \cdot s + \delta_b \cdot \bar{b} - \bar{b}) \\ &= \frac{(1 + \delta_b) \cdot s + (1 - \delta_b) \cdot \bar{b}}{2} \end{aligned} \]

Now, We have found the Optimal Commitment strategies profile for the seller and buyer. The seller proposed a price \(p_1 = \frac{(1 + \delta_b) \cdot s + (1 - \delta_b) \cdot \bar{b}}{2}\). The buyer decide whether it rejects or accepts by a cutoff strategy where it rejects if \(b < \frac{p_1 - \delta_b \cdot x^{-1}(p_1)}{1 - \delta_b}\) and accepts otherwise. Suppose the first proposal is rejected, the buyer proposed a price \(p_2 = x^{-1}(p_1)\). Finally, the buyer decides whether it rejects or accepts by a cutoff strategy where it rejects if \(s < p_2\) and accepts if \(s \geq p_2\).

We can also simplify the formula of \[ b^* = \frac{s + \bar{b}}{2} \qquad u_s(s) = \frac{(1 - \delta_b)(\bar{b} - s)^2}{4(\bar{b} - \underline{b})} . \]

At the end of this section, we consider the case study that shown in the last subsection, where \(\underline{b} = \underline{s} = 0, \bar{s} = \bar{b} = 1, \delta_s = \delta_b = 0.5\). Then seller's initial price should be \[ x(s) = \frac{3}{4}s + \frac{1}{4} . \] Surprisingly, the Optimal Commitment solution is almost same as the one in Sequential Equilibrium (shown in Fig. 2). Apparently, We can simply check that they are indeed different. This example shows that even though OC solution and SE solution have different properties, they can be similar in each step.

Recall that, in SE solution, \(x'(s) = \frac{(\delta_s + \delta_b)(x - s)}{2x - (1 - \delta_b) \cdot \bar{b} - (1 + \delta_b) \cdot s}\). And in OC solution, \(x(s) = \frac{(1 + \delta_b) \cdot s + (1 - \delta_b) \cdot \bar{b}}{2}\). A observation is that the denominator of SE's \(x'(s)\) is 0 pugging in OC's \(x(s)\). So the solving \(x(s)\) for SE is sensitive, and thus, the computation is difficult.

Discussion

We have already shown two reasonable solutions for the two-round bargaining. Although the Optimal Commitment solution has weaker property, we can easily calculate the strategy and get some observation. From the result of the Optimal Commitment, we obverse three aspects of negotiation power, proposing order, patience, and the bound of the opponent's base price.

• Firstly, the side who propose first loses its initiative. In our model, the buyer proposes first and the seller can estimate the exact reserved price. Using the exact reserved price, the buyer is more aggressive than the seller. For example, if the buyer is patient enough(i.e. delay rate \(\delta_b = 1\)), they will trade on the reserved price and the seller can profit \(0\). In real-world negotiation, people avoid proposing first especially when they have not collected enough information about their opponent.
• Secondly, the less patience or lack of time leads to less profit. In our model, we use the decay rate \(\delta_s, \delta_b\) to characterize the impatience or extent of lack of time. If the buyer wants to close the deal sooner, it wants to accept an earlier proposal rather than propose a new one which is more beneficial to it. This corresponds to a smaller decay rate \(\delta_b\). By the formula of \(x(s)\) and \(u_s(s)\), the smaller decay rate \(\delta_b\) results in higher initial price and higher expected utility.
• Thirdly, a high upper bound of the buyer's expected price makes the seller has an incentive to propose a high initial price. Some people say that it is a rule of thumb that the seller always should ask for a high price in its first proposal. However, it is only partially true from our model's perspective. For the seller in our model, the initial value is always higher than its reserved price, while how high it should propose depends on the upper bound of the buyer's expected price \(\bar{b}\). If \(\bar{b}\) is very high, the seller should propose a high price. Conversely, if \(\bar{b}\) is about the same as \(s\), the seller shouldn't propose a high price, and otherwise, the buyer will give up. This result is similar to real-world negotiation. In souvenirs shops, the seller always proposes a high price because the customers have a high expected price. And in fruit and vegetable markets, the seller always proposes a low price because the customers have a expected price near the reserved price.

Conclusion

In this paper, we build a two-round bargaining model with both-side uncertainty to study the negotiation power. We construct a Sequential Equilibrium for this game and characterize it by an ordinary differential equation. Due to the computational difficulty of the SE solution, we further find a feasible but almost the same good solution, Optimal Commitment. Under this solution, we restate the strategy for both buyer and seller and analyze the negotiation power from the model's perspective.

However, there are some limitations in this our work. We only consider two-round bargaining, and the two-round bargaining is too shallow to show the process of general negotiations. For example, the range of seller's price \(\bar{s}, \underline{s}\) and the decay rate \(\delta_s\) almost do not influence the solution, while these parameters should symmetric to the buyer's ones in general negotiations. Moreover, we assume the distribution of the seller's reserved price and the buyer's expected price is common knowledge but this is not the case in reality. Some previous works have pointed out that the seller does not know the buyer's belief about the seller's uncertainty. [2] This means the distribution should be a second-order uncertainty. The last but not least, we do not prove that the Sequential Equilibrium that we construct is the unique one, and we do not have clear ideas about whether SE is a good solution concept for general negotiations.

The highlight of this work is that we successfully established a solvable bargaining model and shows the source of negotiation power. From our model, we obverse the following three factors that influence the negotiation power, proposed order, patience, and the bound of the opponent's base price. Apart from confirming the practical negotiation principles, we also can have a better understanding of how these factors influence the negotiation result.

Reference

[1] Peter C. Cramton. Strategic delay in bargaining with two-sided uncertainty. The Review of Economic Studies, 59(1):205–225, 1992.

[2] Yossi Feinberg and Andrzej Skrzypacz. UNCERTAINTY ABOUT UNCERTAINTY AND DELAY IN BARGAINING. Technical Report 1, 2005.

[3] Bo An, Nicola Gatti, and Victor Lesser. Bilateral Bargaining with One-Sided TwoType Uncertainty. Technical report.