A shared-revenue Bertrand game

Many modern businesses partner with independent service providers to increase efficiency. One method through which this is done is for a company to let a separate company provide services/goods to customers in exchange for a share of the revenue. The independent service provider in turn may benefit from the wider visibility it gains through access to the original company’s pool of customers. This is seen in a number of real-world scenarios, including but not limited to:

1. (A)

   Consider a large store which allows independent sellers to use some of its shelf space to sell products in return for a portion of the selling price. The large store may of course obtain the same products from a vendor and sell them on its own, and the independent seller may set up its own store, but depending on the costs for each party, both parties might benefit from having the independent seller’s products available in the larger store.

2. (B)

   Consider a construction and maintenance company, with a large pool of customers, providing a variety of services such as plumbing, repairs, painting services, etc. The company can either provide a specific service on its own, or refer the customer to a contractor who sets the price for the service, and pays the original company a referral fee.

3. (C)

   Consider a ridesharing or taxi platform. When a passenger requests a ride, they have the option of transporting the passenger with either (i) a self-driving car, owned by the ridesharing platform, or (ii) an independent human driver, whom the ridesharing platform charges a fee for each passenger they refer.

4. (D)

   A large airline could either operate a short regional flight with their own aircraft or allow a regional partner to operate the flight.

5. (E)

   A healthcare provider can either provide specialist services in-house or refer their patients to an external contractor, charging a referral fee.

In this work we provide a game theoretic model for reasoning about the interaction between agents in scenarios such as these with a variation of the Bertrand pricing game (Bertrand, 1883). For concreteness, we will focus our intuition and analysis on scenario (A) from Section 1.1, though our model can explain phenomena in domains far beyond retail sales. In accordance with scenario (A), the “shared-revenue Bertrand game” is played between a retailer, and an independent goods/services provider which we will refer to as the independent seller (seller). Both players sell an identical good/service, with each agent having a different cost. Under the revenue sharing program, whenever the independent seller fulfills demand, a portion of the revenue, called the referral fee, is transferred from the independent seller to the retailer. In Section 2, we set up the game in more detail, then in Section 3 we identify the Nash equilibria of this game. In Section 4, we interpret and analyze the equilibria of the shared-revenue Bertrand game in depth with different costs and fee structures.

The referral fee is a constant parameter of the shared-revenue Bertrand game of Section 2. However, to make the game more realistic, in Section 5 we introduce the “fee optimization game,” which models the referral fee as an action for the retailer that is chosen before playing the shared-revenue Bertrand game. Finally, in Section 6, we introduce the “outside option game”: after observing the referral fee chosen by the retailer, the independent seller can choose to sell on their own instead of through the retailer.

The equilibria of the “fee optimization game” reveal that the retailer always prefers to allow the independent seller to fulfill demand in equilibrium, setting the referral fee strategically so as to maximize their payoff. There is a sweet spot for the optimal fee such that it is low enough to stimulate economic activity while high enough to maximize payoff. In the fee optimization game of Section 5, we still observe some unrealistic equilibria in which the retailer sets such a high fee that the independent seller must fulfill demand at their effective cost (thus achieving no payoff), which leads us to consider the “outside option game” in Section 6. If the independent seller’s threat of selling on their own is credible, the unrealistic equilibria of Section 5 will no longer be observed, as the retailer must set the referral fee lower to ensure the independent seller doesn’t exercise their outside option. Importantly, our analysis demonstrates that the retailer will frequently be incentivized to set a fee under which both the independent seller increases its payoff compared to selling independently, and the price to the customer is decreased relative to the price either the retailer or the independent seller would have set independently. Our results therefore suggest that such a revenue sharing mechanism can not only increase social welfare, but also simultaneously improve outcomes for both players as well as consumers.

In each of the models, we examine if the revenue sharing program increases the retailer’s payoff compared to simply fulfilling demand on their own, and how it affects the end cost to the customer.

Our model is an extension of the standard Bertrand game (Bertrand, 1883) in which two sellers of an identical product with the same cost set their prices independently. As we justify in Appendix B, we consider the regime in which the sellers’ costs are asymmetric, which is a natural extension of the standard Bertrand game, studied for instance in (Blume, 2003; Kartik, 2011; Demuynck, 2019).

One field in which revenue sharing programs as described in Section 1.1 have been studied is in the independent contractor literature. Much of this literature analyzes relationships with independent contractors through the framework of principal agent theory (Coats, 2002; Chang, 2014). The revenue-sharing program can also be seen as a form of “make-or-buy” decision for the retailer (Coase, 1937; Williamson, 1975; Tadelis, 2002). Slightly less obviously, another field in which revenue sharing programs have been studied is optimal taxation theory, where the referral fee can be viewed as the retailer imposing a tax on the independent seller. Viewed in this light, our results about optimal referral fees resonate strongly with optimal taxation theories such as (Laffer, 2004; Stern, 1976; Stiglitz, 1987).

Finally, the idea of an outside option that we introduce in Section 6 is common in negotiation and bargaining theory (Binmore et al., 1989; Osborne and Rubinstein, 1990), which makes sense because the participation of the independent seller in the retailer’s revenue-sharing program can be seen as a negotiation of a contract. In fact, there is even an “outside option principle” (Watson, 2020), which informally states than an outside option must be credible in order to have an effect on the Nash equilibria of the game; our results are consistent with this.

In order for the game to be nontrivial, we must impose the following constraints:

1. (I)

   There exists a price at which both the retailer and independent seller could achieve a nonzero payoff when fulfilling demand ($0<c_{s},c_{r}<1$).

2. (II)

   The seller’s cost is less than the retailer’s cost ($c_{s}<c_{r}$).

3. (III)

   The referral fee takes some, but not all of the independent seller’s revenue ($\alpha\in(0,1)$).

In Appendix B, we formally show that the game degenerates if these are violated, but the importance of each condition should be fairly intuitive. Overall, the revenue-sharing program should be thought of as for products that the retailer could conceivably sell themselves, but the independent seller is able to do so more efficiently.

We now introduce several important prices and corresponding payoffs which will be used throughout this paper. We also include a full notation table in Appendix A, and include derivations of the key prices and fees in Appendix D.

As we will later prove in Proposition 1, we only need to focus on cases where either the retailer or the independent seller fulfill all demand, and can ignore any split market scenario. When we write a payoff $\pi$ with two subscripts, the first denotes the the player whose payoff we refer to, and the second denotes the player who fulfills all demand. Thus, $\pi_{r,r}(p_{r})$ and $\pi_{s,r}(p_{r})$ are the retailer’s and the independent seller’s payoffs when the retailer fulfills all the demand, respectively. Similarly $\pi_{r,s}(p_{s})$ and $\pi_{s,s}(p_{s})$ are the retailer’s and the independent seller’s payoffs when the independent seller fulfills all the demand, respectively. Explicitly, it follows from (1), (2), and (5) that

In Figure 1 we plot the payoffs of the players as a function of price for two choices of game parameters $(c_{r},c_{s},\alpha)$. Solid and dashed lines denote the players’ payoff when the retailer or independent seller fulfill all demand, respectively, while blue and red lines denote the payoffs to the retailer and the independent seller, respectively.

Figure 1 also illustrates five important price points and their corresponding payoffs which will be used throughout this paper. We denote by $p_{r^{*}}$ and $p_{s^{*}}$ the prices each player would set to maximize their own payoff if they fulfilled all demand, and call these the retailer’s or independent seller’s optimal selling price. Differentiating the players’ single agent payoffs $\pi_{r,r}$ and $\pi_{s,s}$ and equating to zero in (36) and (40) gives the values of these prices and their corresponding payoffs –

We denote by $p_{r_{\text{ind}}}$ and $p_{s_{\text{ind}}}$ the prices at which the retailer and the independent seller are indifferent to which player fulfills demand, i.e. $\pi_{r,r}(p_{r_{\text{ind}}})=\pi_{r,s}(p_{r_{\text{ind}}})$ and $\pi_{s,r}(p_{s_{\text{ind}}})=\pi_{s,s}(p_{s_{\text{ind}}})$. Solving these equations in (38) and (42) yields the following prices and payoffs –

We first note that $\pi_{s,s}(p_{s_{\text{ind}}})=0$, and therefore the independent seller’s indifference price is also its breakeven price.

More interestingly, while the retailer’s breakeven price is still its cost, $c_{r}$, its indifference price is scaled by $(1-\alpha)^{-1}$, but at that price its payoff is nonzero. This is the main feature which differentiates the shared-revenue Bertrand game from the classical Bertrand game, as the retailer can choose to fulfill demand on its own or allow the independent seller to fulfill demand, making revenue from the fee. As a result, this game gives rise to a more complex equilibrium structure which depends on the fees and costs of the players. Note also that because we assume $c_{s}<c_{r}$ (Section 2.1), we have $p_{s_{\text{ind}}}<p_{r_{\text{ind}}}$.

Next, we introduce the price at which the retailer’s payoff when the independent seller fulfills demand is equal to its optimal payoff had it fulfilled demand on its own, i.e.

$p^{\dagger}$ is the solution of a quadratic equation which we derive in (58), and we define it as the larger of the two possible solutions. Furthermore, $p^{\dagger}$ is only well-defined when $\alpha\geq(1-c_{r})^{2}$, and therefore we only show it in the right subplot of Figure 1.

For simplicity, we will only consider pure strategies in this paper; thus, a strategy profile can be characterized fully by a tuple $(p_{r},p_{s})\in\mathcal{S}_{p}\equiv(0,1)\times(0,1)$ denoting the prices chosen by the retailer and independent seller. We first prove, in Proposition 1, that in all equilibria, one player takes the entire market.

While this result is not surprising given that in nearly all standard non-symmetric game theory models, in equilibrium only one player fulfills all demand, this result is clearly not expected to hold in the real world. Game theory results may mimic the world more closely and arrive at more realistic equilibria in which players split demand by modeling factors such as incomplete information and brand loyalty, but these are beyond the scope of this work.

In light of Proposition 1, we can denote all equilibria in the game by either unequal prices $(p_{r},p_{s})\in\mathcal{S}_{p}$ or a shared price $(p,x)\in(0,1)\times\{r,s\}$, where $x$ denotes the player who fulfills the demand. In this subsection, we aim to provide an intuitive sketch of the Nash equilibria.

To narrow our focus, let’s consider shared-price equilibria $(p,s)$ where the independent seller fulfills demand at price $p$ for now. In order for $(p,s)$ to be an Nash equilibrium, the following conditions must be satisfied:

1. (1)

   0 = $\pi_{s,r}(p)\leq\pi_{s,s}(p)$

2. (2)

   $\sup_{p^{\prime}\in[0,p)}\left[\pi_{s,s}(p^{\prime})\right]\leq\pi_{s,s}(p)$

3. (3)

   $\sup_{p^{\prime}\in[0,p)}\left[\pi_{r,r}(p^{\prime})\right]\leq\pi_{r,s}(p)$

If condition 1 doesn’t hold, the independent seller can achieve a higher payoff by increasing its price and letting the retailer fulfill demand. If condition 2 doesn’t hold, the independent seller can increase its payoff by lowering their price and continuing to fulfill all demand. If condition 3 doesn’t hold, then there is some price lower than $p$ the retailer can set to take the entire market and increase its payoff.

In Figure 2, we highlight in yellow the Nash equilibrium prices where the independent seller fulfills demand. In the left subplot of Figure 2, the Nash equilibria occur at prices $p\in[p_{s_{\text{ind}}},p_{r_{\text{ind}}}]$, which satisfies all of the Nash equilibrium conditions:

1. (1)

   The seller achieves nonnegative payoff because $p\geq p_{s_{\text{ind}}}$.

2. (2)

   The independent seller’s payoff would decrease if they set a price $p^{\prime}<p$ because $p\leq p_{s^{*}}$.

3. (3)

   The retailer would not prefer to fulfill demand themselves because $p\leq p_{r_{\text{ind}}}$.

In the right subplot of Figure 2, the Nash equilibria occur at prices $p\in[p_{s_{\text{ind}}},p^{\dagger}]$. Just as in the left subplot, conditions 1 and 2 are satisfied because $p\geq p_{s_{\text{ind}}}$ and $p\leq p_{s^{*}}$. Condition 3 is satisfied because $p\leq p^{\dagger}$. Notice that for the payoff curves in the right subplot, this is a stronger condition than $p\leq p_{r_{\text{ind}}}$. This stronger condition is necessary because if $p>p^{\dagger}$, the retailer would prefer to set price $p_{r^{*}}<p$ and achieve a payoff of

In Appendix D.2, we carry forth this intuition and translate conditions 1-3 into feasible intervals of prices for any $\alpha\in(0,1)$ rather than just for the specific referral fees in Figure 2. For $(p,s)$ to be a Nash equilibrium, we find that the price must satisfy

We acknowledge that it’s still not clear what the equilibrium interval of prices look like for each possible choice of $\alpha$, namely what lower and upper bounds are attained for each possible referral fee – this warrants a longer discussion which we defer to Section 3.3. For instance, $p^{\dagger}$ is not well-defined for small $\alpha$, but we will soon see in (20) that whenever $\left[\max\{p_{r^{*}},p_{s_{\text{ind}}}\},\min\{p^{\dagger},p_{s^{*}}\}\right]$ is nonempty, $\alpha$ is high enough for $p^{\dagger}$ to be well-defined. For now, simply keep in mind that (13) is sufficient only for shared-price Nash equilibria where the independent seller fulfills demand.

In Appendix D.1, we derive the conditions analogous to (13) for shared-price equilibria $(p,r)$ where the retailer fulfills demand and find the following:

Upon further inspection of (14), we notice a subtle problem. Recall that $p_{s_{\text{ind}}}<p_{r_{\text{ind}}}$ always because $c_{s}<c_{r}$. Thus, the interval $\left[p_{r_{\text{ind}}},\min\{p_{s_{\text{ind}}},p_{r^{*}},p_{s^{*}}\}\right]$ is empty for all $\alpha$, which implies there are no shared price-equilibria where the retailer fulfills demand!

This completes the first step of our analysis for shared-price equilibria, but we also need to consider unequal-price equilibria. Indeed, we consider equilibria where $p_{r}<p_{s}$ in Appendix D.3 and equilibria where $p_{s}<p_{r}$ in Appendix D.4. Interested readers can find conditions similar to (13) and (14) in the corresponding appendices; we also include a summary of all Nash equilibria with both equal and unequal prices in Section 3.4.

In this section, we answer the aforementioned question of what the interval in (13) looks like for different values of $\alpha$. To do this, we will introduce five threshold fees that delineate the transitions between when different Nash equilibria can occur (skip ahead to Figure 3 for a preview). Similar analysis as in this section for unequal price equilibria can be found in Appendices D.3 and D.4.

Consider first the left interval of (13), $p\in\left[p_{s_{\text{ind}}},\min\{p_{r_{\text{ind}}},p_{r^{*}},p_{s^{*}}\}\right]$. We know that $p_{s_{\text{ind}}}<p_{r_{\text{ind}}}$ always. We have $p_{s_{\text{ind}}}\leq p_{s^{*}}=\frac{1}{2}(1+p_{s_{\text{ind}}})$ so long as $p_{s_{\text{ind}}}\leq 1$, which occurs when $\alpha\leq 1-c_{s}$. The condition for $p_{s_{\text{ind}}}\leq p_{r^{*}}$ is not quite as clear, so we introduce the retailer optimal price feasibility fee $\alpha_{r^{*}}$ (derived in (46)) –

The name comes from the fact that $p_{r^{*}}\leq p_{s_{\text{ind}}}$ is necessary for any equilibrium where the retailer fulfills demand at their optimal price, which we prove in Appendix D.3. Note also that $\alpha_{r^{*}}\leq 1-c_{s}$, which we show in (52). Thus, the left interval of (13) is nonempty if and only if $\alpha\leq\alpha_{r^{*}}$.

Now, we turn our attention to the value of $\min\{p_{r_{\text{ind}}},p_{r^{*}},p_{s^{*}}\}$, the upper bound of the left interval of (13). This motivates the optimality switching fee $\alpha_{s^{*},r^{*}}$ (derived in (52)), the seller optimal price feasibility fee $\alpha_{s^{*}}$ (derived in (50)), and the retailer $p^{\dagger}$ relevance fee $\alpha^{\dagger}_{r}$ (derived in (49)) –

Analogously to $\alpha_{r^{*}}$, we will soon find that $\alpha\geq\alpha_{s^{*}}$ is necessary for an equilibrium where the independent seller fulfills demand at $p_{s^{*}}$. Intersecting the inequalities in (15), (16), (17), and (18) is sufficient to determine what the left interval of (13) looks like in each region of parameter space. We will omit the algebra here and instead opt to summarize this in Section 3.4. However, throughout the course of the derivation in Appendix D, an important condition emerges that determines the relative ordering of $\alpha_{s^{*},r^{*}}$, $\alpha_{s^{*}}$, and $\alpha^{\dagger}_{r}$. We show in (54) and (55) that this seller optimal feasibility cost $c_{s^{*}}$ satisfies the following –

Finally, let’s turn our attention to the right interval of (13), $p\in\left[\max\{p_{r^{*}},p_{s_{\text{ind}}}\},\min\{p^{\dagger},p_{s^{*}}\}\right]$. Fortunately, we have already determined the relative ordering of each of the prices in the right interval, except for $p^{\dagger}$. For these conditions involving $p^{\dagger}$, we introduce the $p^{\dagger}$ relevance fees for the retailer and seller (derived in (59) and (62)):

Note that we already introduced $\alpha^{\dagger}_{r}$ in (18). The notation arises because we will show that $\alpha^{\dagger}_{r}\leq\alpha\leq\alpha^{\dagger}_{s}$ is necessary for any equilibrium where demand is fulfilled at $p^{\dagger}$. We don’t include a closed form here for $\alpha^{\dagger}_{s}$ because it is uninsightful, but we remark that $\alpha^{\dagger}_{s}\in[\alpha_{r^{*}},1-c_{s}]$ and that a closed form is in (62). Finally, the relative ordering of $p_{s^{*}}$ and $p^{\dagger}$ is complex, and there unfortunately is no special $\hat{\alpha}$ for which $p_{s^{*}}\leq p^{\dagger}\iff\alpha\leq\hat{\alpha}$. Thus, we will leave that condition unsimplified for now.

Considering all possible values of $\alpha$ and all possible price configurations, we summarize the Nash equilibria of the game below. We denote sets of equilibrium price configurations as $(S_{r},S_{s})\subset\mathcal{S}_{p}$ and singletons without the set notation, specifically $x\equiv\{x\}$. Recall there are no shared-price equilibria where the retailer fulfills demand, thus all sets of shared-price equilibria $(S_{r},S_{s})=(p_{s},[\cdot,\cdot])$ implicitly assume that the independent seller is fulfilling demand at price $p_{s}\in[\cdot,\cdot]$, not the retailer.

Notation
We denote Nash equilibrium strategies by price configurations $(p_{r},p_{s})\in(S_{r},S_{s})\subset\mathcal{S}_{p}$.

The admissibility criterion allows us to rule out any weakly dominated strategies for the retailer and independent seller. For instance, in the left subplot of Figure 2, all Nash equilibrium prices $p_{r}^{\prime}\in[p_{s_{\text{ind}}},p_{r_{\text{ind}}})$ are weakly dominated by $p_{r_{\text{ind}}}$ for the retailer. Indeed, for all $p_{r}^{\prime}<p_{r_{\text{ind}}}$, we have $\pi_{r,s}(p_{r}^{\prime})>\pi_{r,r}(p_{r}^{\prime})$ by construction of $p_{r_{\text{ind}}}$. Thus, if the seller sets a price $p_{s}\in(p_{r}^{\prime},p_{r_{\text{ind}}})$, then

which implies that the retailer would have preferred to set price $p_{r_{\text{ind}}}$ rather than $p_{r}^{\prime}$. Thus, the only admissible Nash equilibrium in the left subplot of Figure 2 is $p=p_{r_{\text{ind}}}$.

In Appendix E.1, we more thoroughly consider the retailer and independent seller’s strategies, and find the strategies that are not weakly dominated are

The intuition for the independent seller and the retailer is the same – we can eliminate all prices that are not between their indifference and optimal prices – however, for the retailer we do not always have that $p_{r_{\text{ind}}}\leq p_{r^{*}}$. Given the admissible strategies in (25), we can simply intersect the sets of admissible prices with the Nash equilibrium price configurations we found in (22) and (23). For instance, in the right subplot of Figure 2, we find that the admissible Nash equilibria are

We perform these intersections throughout all of parameter space in Appendix E.1, and include a complete summary of admissible equilibria in (70), (71), and (72).

The relative Pareto optimality criterion allows us to further eliminate one admissible equilibrium, which occurs in the following case:

The equilibrium where the retailer fulfills demand at $p_{r^{*}}$ is Pareto suboptimal relative to the shared-price equilibrium $p^{\dagger}$ where the independent seller fulfills demand. Indeed, by construction of $p^{\dagger}$, in both cases the retailer achieves the same payoff of $\pi_{r,r}(p_{r^{*}})=\pi_{r,s}(p^{\dagger})$ so they are indifferent. However, because $\alpha\leq\alpha^{\dagger}_{s}$ implies that $p_{s_{\text{ind}}}\leq p^{\dagger}$, the independent seller achieves a positive payoff when fulfilling demand at $p^{\dagger}$ while they achieve a payoff of $0$ when allowing the retailer to fulfill demand at $p_{r^{*}}$. Thus, they strictly prefer fulfilling demand at $p^{\dagger}$ rather than allowing the retailer to fulfill demand at $p_{r^{*}}$.

As a remark, we note that a reasonable argument could be made for preferring the equilibrium where the retailer fulfills demand at $p_{r^{*}}$. In this case $p_{r^{*}}<p^{\dagger}$, so despite the Pareto optimality argument, the customer will be able to buy the product at a lower price if the retailer fulfills demand at $p_{r^{*}}$ instead of the independent seller fulfilling demand at $p^{\dagger}$. With this lowest price criterion, $p_{r^{*}}$ would in fact be the most likely equilibrium in this case, as $p_{r^{*}}<p_{s_{\text{ind}}}$ as well. However, whatever we choose does not substantially affect the results, and the relative Pareto optimality criteria will continue to be sensible for us as we generalize the game, thus we proceed with this assumption.

Even after applying these two refinements, inspecting (70), (71), and (72) we can still see that we are left with many intervals of prices. However, in order to interpret the real-world implications of the equilibria, it is prudent to consider the equilibrium outcomes: at what price the customer will purchase the product and from whom they will buy. Indeed, letting the prices be $p\in P\subset[0,1]$ and the fulfiller of demand be $x\in\{r,s\}$, we find the following equilibria:

Notation
We denote an equilibrium outcome by $(P,x)\subset(0,1)\times\{r,s\}$, indicating that there is an admissible, Pareto optimal Nash equilibrium where player $x$ fulfills demand at any price $p\in P\subset(0,1)$.

A quick glance at (26) and (27) shows that when $\alpha$ is sufficiently high, the independent seller is “priced out” of the market and the retailer will fulfill demand at their single agent price as if the independent seller did not exist. When $\alpha$ is small, the independent seller sells at the retailer’s indifference price, illustrating that they have to prioritize preventing the retailer from undercutting them over maximizing their own payoff. However, when $\alpha$ is in a certain sweet spot, the independent seller may be able to sell at their optimal price in equilibrium. We will see soon that everybody wins in this regime – if the retailer gets to choose the referral fee, they would prefer to choose a referral fee in this sweet spot rather than one that is too high or too low.

Our results in (26) and (27) show that given a point in the parameter space $(c_{r},c_{s},\alpha)$, we can predict which player will fulfill demand and the price(s) at which they will fulfill demand in an admissible, Pareto optimal Nash equilibrium. In this section, we build intuition about how the parameter space is divided into regions based on these possible equilibria and discuss the business implications for the retailer, both in terms of the retailer’s payoff and in terms of the effect on the prices. Importantly, we will contrast these metrics with the alternative of not having the revenue sharing option available to outside sellers – i.e. compare them with the price the retailer would have set and its payoff in the absence of an independent seller.

In Figure 3 we plot two cross sections of the parameter space, one plotting $c_{r}$ vs $\alpha$ at a fixed $c_{s}=0.4$ and one with $c_{s}$ vs $\alpha$ at a fixed $c_{r}=0.6$. The different colors correspond to the admissible, Pareto optimal equilibrium for each set of parameters, with the “continuum” corresponding to the interval of prices $\left(p\in\left[\max\left\{p_{r^{*}},p_{s_{\text{ind}}}\right\},\min\left\{p_{s^{*}},p^{\dagger}\right\}\right],s\right)$.

Notice that when $c_{s}\geq c_{s^{*}}$ there is no value of $\alpha$ that yields an equilibrium $(p_{s^{*}},s)$ in the red region – this is where the $c_{s^{*}}$ notation comes from. Additionally, notice that we may have that $\alpha_{s^{*}}<0$, in which case the green region is not feasible as we cannot have $\alpha\leq\alpha_{s^{*}}<0$.

We will impose the same refinements on the Nash equilibria of the sequential game as we did in Section 4 – namely, that they must be admissible and Pareto optimal relative to other equilibria. Admissibility has no effect on the equilibria of the sequential game, and Pareto optimality yields us the natural refinement that the retailer will choose the minimum $\alpha$ necessary to attain their maximum possible equilibrium payoff. More formal justification can be found in Appendix F.1.

We will impose the further refinement of subgame perfection (Rasmusen, 1989), which is equivalent to backward induction in this game of perfect, complete information. In order to achieve this, the retailer and independent seller must choose a price configuration that is an admissible, Pareto optimal equilibrium of the shared-revenue Bertrand subgame. However, there are many choices of such price configurations. We will thus define an equilibrium strategy profile $\rho$ for the subgame. Formally, we require that $\rho$ is a strategy profile with the property that for every possible $\alpha,c_{r},c_{s}$, $\rho(\alpha,c_{r},c_{s})\in\mathcal{S}_{p}$ is an admissible, Pareto optimal equilibrium price configuration of the shared-revenue Bertrand game as outlined in (70), (71), and (72)⁵⁵5This function $\rho$ is necessary for subgame perfection and could come from anywhere. Intuitively, we could think of the retailer and independent seller agreeing in advance on what prices they will set for each choice of $\alpha$, given common knowledge of their prices $c_{r},c_{s}$. We could also think of the retailer as more powerful and give them the ability to choose $\rho$, requiring that the independent seller agree to choose the prices outlined in $\rho$ as a condition of the revenue sharing program..

To give some concrete examples, some natural choices for $\rho$ are:

1. (I)

   $\underline{\rho}(\alpha)$ – for each $\alpha$, $\underline{\rho}(\alpha)\in\mathcal{S}_{p}$ is an admissible, Pareto optimal price configuration where the equilibrium outcome has demand being fulfilled at the minimum possible price

2. (II)

   $\overline{\rho}(\alpha)$ – for each $\alpha$, $\overline{\rho}(\alpha)\in\mathcal{S}_{p}$ is an admissible, Pareto optimal price configuration where the equilibrium outcome has demand being fulfilled at the maximum possible price

By inspection of (26) and (27), the only region in which $\rho$ makes a nontrivial choice is in the gray continuum region of Figure 3, which allows demand to be fulfilled at $p\in\left[\max\left\{p_{r^{*}},p_{s_{\text{ind}}}\right\},\min\left\{p_{s^{*}},p^{\dagger}\right\}\right]$ when $\alpha\in\left[\max\left\{\alpha^{\dagger}_{r},\alpha_{s^{*},r^{*}}\right\},\alpha^{\dagger}_{s}\right]$. Continuing our examples, we would have that

1. (I)

   For all $\alpha\in\left[\max\left\{\alpha^{\dagger}_{r},\alpha_{s^{*},r^{*}}\right\},\alpha^{\dagger}_{s}\right]$, $\underline{\rho}(\alpha)\equiv\left(\max\left\{p_{r^{*}},p_{s_{\text{ind}}}\right\},\max\left\{p_{r^{*}},p_{s_{\text{ind}}}\right\}\right)\in\mathcal{S}_{p}$

2. (II)

   For all $\alpha\in\left[\max\left\{\alpha^{\dagger}_{r},\alpha_{s^{*},r^{*}}\right\},\alpha^{\dagger}_{s}\right]$, $\overline{\rho}(\alpha)\equiv\left(\min\left\{p_{s^{*}},p^{\dagger}\right\},\min\left\{p_{s^{*}},p^{\dagger}\right\}\right)\in\mathcal{S}_{p}$

For all $\alpha\notin\left[\max\left\{\alpha^{\dagger}_{r},\alpha_{s^{*},r^{*}}\right\},\alpha^{\dagger}_{s}\right]$, there is only one admissible, Pareto optimal equilibrium outcome as specified by (26) and (27), thus we will always observe the same outcome irrespective of our choice of $\rho$. For any $\rho$, the subgame perfect, admissible, Pareto optimal equilibria are as follows:

Given
$\rho(\alpha,c_{r},c_{s})\in\mathcal{S}_{p}$ – an admissible, Pareto optimal equilibrium strategy profile of the shared revenue Bertrand game for every set of parameters

To complete our analysis, our natural next step is to find $\alpha_{*}$. We’ll denote as $\pi_{r}^{\text{(eq)}}(\rho(\alpha),\alpha,c_{r},c_{s})$ the payoff function in equilibrium of the shared-revenue Bertrand game for the retailer with strategy $\rho$ and $\pi_{s}^{\text{(eq)}}$ the analogous function for the independent seller. With this notation, it’s clear that

However, the functional dependence of $\alpha_{*}$ on $\rho$ makes this optimization problem complicated in general. To build intuition, we will begin by considering only $\underline{\rho}$ and $\overline{\rho}$. In Appendix F.4, we show the simple result that

In other words, if the retailer and independent seller agree to play strategy $\underline{\rho}$, the retailer will always choose referral fee $\underline{\alpha}=\alpha_{r^{*}}$ in equilibrium of the Fee Optimization Game.

The optimal fee for $\overline{\rho}$ is slightly more complicated than for $\underline{\rho}$, but we show in Appendix F.3 that

To illustrate $\underline{\alpha}$ and $\overline{\alpha}$, we plot typical payoff curves in Figure 4. The background of the plot corresponds to the equilibria in Figure 3. In the left plot, there does exist an $\alpha$ such that $\pi_{r,s}(p_{s^{*}},\alpha)>\pi_{r,r}(p_{r^{*}})$, so $\alpha_{*}(\overline{\rho})=\overline{\alpha}$. In the right plot, for all $\alpha$ we have $\pi_{r,s}(p_{s^{*}},\alpha)\leq\pi_{r,r}(p_{r^{*}})$, so $\alpha_{*}(\overline{\rho})=\alpha^{\dagger}_{r}$.

In Section 5.2, we interpreted the equilibria of the Fee Optimization Game under specifically the equilibrium strategy profiles $\underline{\rho}$ and $\overline{\rho}$. However, as a result of monotonicity of the payoff functions in the gray region (the only region where the choice of $\rho$ is nontrivial), we can bound the payoffs for any $\rho$ by the payoffs for $\underline{\rho}$ and $\overline{\rho}$. Thus, the shaded areas in the gray region of Figure 4 correspond to the range of possible payoffs for the retailer and seller, with the exact values depending on the choice of $\rho$. Slightly more formally, in Appendix F.2 we prove the following:

In light of Corollary 1, we can bound the equilibrium payoff of the Fee Optimization Game $\pi_{r}$ for any strategy profile $\rho$ –

Bounds on $\pi_{s}$ and $\alpha_{*}$ follow similarly by considering $\pi_{r}^{\text{(eq)}}$ under the strategy profiles $\underline{\rho}$ and $\overline{\rho}$. Indeed, we show in Appendix F.3 and F.4 that for any equilibrium strategy profile $\rho$, the admissible, Pareto optimal, subgame perfect Nash equilibria of the sequential game satisfy:

One important takeaway from (30) and (31) is that the equilibrium $\alpha_{*}$ is bounded away from $1$. The existence of such an optimal fee is echoed by optimal taxation theories such as the Laffer curve (Laffer, 2004) – it is not in the retailer’s interest to take all the independent seller’s revenue.

In light of this general result, our analysis from Section 5.2 has far-reaching implications. Importantly, examining Figure 4, we see that the retailer will always induce an equilibrium where the independent seller fulfills demand at some $p\in\left[\max\left\{p_{s_{\text{ind}}},p_{r^{*}}\right\},\left\{p_{s^{*}},p^{\dagger}\right\}\right]$ depending on $\rho$ (red or gray regions). Interestingly, the socially suboptimal green region where the independent seller fulfills demand at $p_{r_{\text{ind}}}$ is never observed in equilibrium of the Fee Optimization Game.

We do also find that the retailer never sets the fee high enough such that they fulfill demand themselves at price $p_{r^{*}}$ (blue region). However, this is less interesting, because it is an artifact of the relative Pareto optimality criteria that we applied when refining equilibria in Section 4.2; we argued there that the retailer would prefer for the independent seller to fulfill demand at $p^{\dagger}$ rather fulfilling demand themselves at $p_{r^{*}}$.

To finish defining the Outside Option Game, we must outline the structure of this “leaving” subgame. When not clear from context, we will use superscripts $\cdot^{(\ell)}$ to denote relevant quantities in the leaving subgame and $\cdot^{(o)}$ to denote relevant quantities in the outside option game as a whole. There are many reasonable specifications for this leaving subgame, and more sophisticated models may better capture reality, but as a first step we choose a very simple model: if the independent seller chooses to leave, they will play a standard Bertrand game against the retailer.

With the constraint that $\alpha>0$, it seems obvious that the independent seller would always leave to avoid paying the referral fee. Thus, we will posit that if they choose to sell on their own, the effective cost to the independent seller will be $c_{s}+\delta$. This additive $\delta$ factor could represent any number of costs that the independent seller might incur if selling without the retailer such as advertising, shipping, or storing inventory. It could even represent an opportunity cost that the smaller independent seller incurs by not exposing their product to the larger retailer’s customers.

Whatever it represents, we will assume that there exists a $\delta>0$ such that the independent seller does not always prefer to leave. We now have a standard Bertrand game with potentially asymmetric costs depending on the value of $\delta$, for which the equilibria have been derived in (Blume, 2003; Kartik, 2011) for instance. However, if $\delta\geq c_{r}-c_{s}$ such that $c_{s}+\delta\geq c_{r}$, the independent seller achieves an equilibrium payoff of $0$ if they exercise their outside option, making leaving the revenue sharing program a non-credible threat. This can be seen by adapting the derivation in Appendix G to one where $c_{r}\leq c_{s}+\delta$ instead of $c_{s}+\delta<c_{r}$. Thus, with the constraint that $\delta\in(0,c_{r}-c_{s})$, we show in Appendix G that the admissible equilibrium outcomes of the leaving subgame are