A Case for Percentage Commission Contracts: The Impact of a “Race” Among Agents (P.6)

Agent Competition

Now we consider the implications of multiple agents who compete to procure a buyer for the seller’s property. Suppose there are n agents, and the first agent to procure a buyer is awarded the full commission, αP. Agents are identical in that each agent has an identical cost of effort intensity function, c(λ), and reservation payoff, φ.

We assume that each agent participates in the sale of every listing on the market. To highlight our objective in the simplest manner possible, we focus on a representative listing and study the competition among agents for that listing. To keep the analysis simple, we assume no interactions of effort levels across agents and across properties. These simplifications enable us to focus on the competition among agents for a representative listing.9

The seller gains access to the MLS system (by paying a fixed fee, say) and offers a commission to the agent who first brings a ready, willing and able buyer. By allowing the seller to have access to the MLS system, we eliminate the search by agents for new listings. Recall that competition among agents for new listings is crucial for the first-best outcome obtained in Williams (1998). We will prove here that the percentage commission system can yield first-best outcome (even) in the absence of this competition for new listings. This distinction is very significant because time spent for new clients has a private value to each agent but no value to their clients. Thus, we show that this wasteful activity is not necessary for the percentage commission system to be efficient. On the contrary, competition for new listings in our model will be an impediment to achieving the first-best outcome.

Each agent searches continuously until a buyer is found by one of them. Although a strategy for each broker is to determine the optimal search intensity time path, we only consider the non-cooperative Nash equilibrium of stationary pure strategies. With n agents, the stochastic time X until a buyer is found for a particular seller is distributed as $$X \sim \exp \left( {\sum\limits_j {\lambda _j } } \right)$$ with $$E\left( X \right) = \frac{1}{{\sum\limits_j {\lambda _j } }}$$, and $$\sum\limits_j {\lambda _j } dt$$ is the probability that a buyer is found in the short time interval dt.10

Given a sale at time t, the payoff for the seller found in Eq. 1 remains unchanged, but the payoff for agent i is now conditional on the fact that she procures the buyer,

$$\pi _i = \alpha Pe^{ - rt} \left( {\frac{{\lambda _i }}{{\sum\limits_j {\lambda _j } }}} \right) - \frac{{c\left( {\lambda _i } \right)}}{r}\left( {1 - e^{ - rt} } \right),$$
(9)

where $$\frac{{\lambda _i }}{{\sum\limits_j {\lambda _j } }}dt$$ is the probability that agent i is the first agent to procure a buyer in the short time interval dt.

With competition among agents, the present value of the unconditional expected payoff to the seller becomes,

$$\begin{array}{*{20}l}   {{V{\left( {\alpha ,\lambda _{1}  \ldots \lambda _{n} } \right)} = {\int\limits_0^\infty  {\pi _{s} {\sum\limits_j {\lambda _{j} } }e^{{ - {\sum\limits_j {\lambda _{j} } }t}} dt} },\quad or} \hfill}  \\   {{V{\left( {\alpha ,\lambda _{1}  \ldots \lambda _{n} } \right)} = \frac{{{\left( {1 - \alpha } \right)}P{\sum\limits_j {\lambda _{j} } }}}{{r + {\sum\limits_j {\lambda _{j} } }}}.} \hfill}  \\ \end{array} $$
(10)
The present value of the unconditional expected payoff to agent i is,

$$\begin{array}{*{20}l}   {{U_{i} {\left( {\alpha ,\lambda _{1}  \ldots \lambda _{n} } \right)} = {\int\limits_0^\infty  {\pi _{i} {\sum\limits_j {\lambda _{j} } }e^{{ - {\sum\limits_j {\lambda _{j} } }t}} dt} },\quad or} \hfill}  \\   {{U_{i} {\left( {\alpha ,\lambda _{1}  \ldots \lambda _{n} } \right)} = \frac{{\alpha P\lambda _{i}  - c{\left( {\lambda _{i} } \right)}}}{{r + {\sum\limits_j {\lambda _{j} } }}}.} \hfill}  \\ \end{array} $$
(11)
Each agent chooses his intensity level, λ*, based on his expectation about other agents’ levels of effort intensity. A non-cooperative solution to this game is a vector of intensity levels $$\lambda _1^* ,\lambda _2^* , \ldots \lambda _n^* $$ such that $$\lambda _i^* $$ maximizes Eq. 11. The solution vector is given by the FOCs,

$$\alpha P - U\left( {\alpha ,\lambda _1^ * , \ldots \lambda _n^ * } \right) = c\prime \left( {\lambda _i^ * } \right)\quad for\quad i = 1, \ldots ,n.$$
(12)
Observe that agent i’s utility, U i , is adversely affected from an increase in intensity by agent j,

$$\frac{{\partial U_i }}{{\partial \lambda _j }} = - \frac{{\alpha P\lambda _i - c\left( {\lambda _i } \right)}}{{\left( {r + \sum\limits_j {\lambda _j } } \right)^2 }} = - \frac{{U_i }}{{r + \sum\limits_j {\lambda _j } }} <0.$$
(12a)

In other words, an increase in any agent’s intensity level imposes negative externalities on other agents. This is the basic insight from Mortensen (1982): The compensation rule which allocates a payoff to the first agent to procure a buyer induces a race in which each agent exerts additional intensity in order to improve the probability that he or she is the procuring agent. The outcome of these negative externalities is that each agent expends too much intensity to sell the seller’s asset. In other words, there exist vectors of intensity levels that are smaller than the equilibrium vector that results from Eq. 12 and are preferred by all the agents. Mortensen (1982) proposes a solution to internalize these externalities by modifying the property rights such that the procuring agent compensates the other agents for the lost value of continued search.

However, the game here differs from the race game in Mortensen (1982) in an important way. In addition to the race among the agents, the problem here also involves a principal-agent relationship between the seller and the agents. Thus, the concern here is not only the impact of an agent’s intensity choice on other agents’ payoffs but also its impact on the principal’s payoff.

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