3 myoptimal brokers

Figure 5 shows the price dynamics for the system just described in the case where the three brokers are myoptimal. The set of possible prices is quantized in increments of 0.002, and each broker performs an exhaustive search among the 4008 possible states. If a consumer perceives two brokers to be equally attractive, the broker with the lower index is preferred.

We now follow the dynamics, starting from an initial configuration in which each broker is in the state (0.480,111) (i.e. each has price and is interested in all three categories). In the simulation run depicted in Fig. 5, broker 3 moves first, and chooses to set its state to (0.560,100). Broker 2 follows, choosing (0.564,010). Broker 1 is selected next. By choosing (0.586,001), broker 1 would make a profit . However, the random generation of consumer interests yields a very slight bias in favor of category 1, and it turns out that broker 1 can do even better ( ) by choosing (0.560,100), undercutting broker 3 and triggering a price war over the niche (100). Meanwhile, broker 2, in the absence of any other competition for category 2, increases its price to the optimal single-category-monopoly value: (0.584,010). Note that this is very close to the price that optimizes in a system with an infinite number of consumers, as computed from Eq. 12 in Appendix A: .

  
Figure 5: Price and niche war timeseries: vs. t for 3 myoptimal brokers and 3 categories, with , V=1. See text for other parameters.

  
Figure 6: Profit for broker 2 as a function of time; same simulation run as in Fig. 5.

  
Figure 7: Sum over all consumer's utilities and total number of subscriptions; same simulation run as in Fig. 5.

Now the high price for category 2 increases its attractiveness, and broker 3 immediately gives up its fight over (100) with broker 1, and now undercuts broker 2 with (0.582,010). With brokers 2 and 3 now specializing in category 2, broker 1 finds it most profitable to offer both categories 1 and 3: (0.564,101). Immediately thereafter, all three brokers join in a price war over the 101 configuration, during which the price is ultimately driven down to 0.540. Now it becomes most profitable to specialize purely in category 2, with price 0.584 (0.584,010). Immediately, a second broker joins into the battle over category 2, causing the remaining broker at (0.540,101) to raise its price, resulting in (0.564,101), instigating yet another price war over the 101 configuration. Although the stochasticity of the order in which brokers make decisions causes some variation in the exact details, the price war cycle continues in this fashion indefinitely.

In summary, after a short initial transient, the system alternates between two price wars: a short-lived one between two brokers vying for the 010 configuration, and a longer-lived one in which all three brokers vie for the 101 configuration. A broker participating in the 010 price war receives its expected profit when it undercuts its competitor, and zero when it is being undercut. During the long 101 price war, a broker will be undercut two thirds of the time, and will thus receive just one third of what it expects. This is illustrated in Fig. 6, which tracks the profit of broker 2 as a function of time.

Price wars are clearly harmful to brokers. In this particular model, they hurt the consumers as well, as illustrated in Fig. 7. During the 010 price war, a single broker is left to offer both categories 1 and 3, which is unsatisfactory to consumers who are highly interested only in one of the two categories. During the long 101 price war, category 2 is completely unavailable to consumers, so the total consumer utility is even lower during this phase than during the 010 price war. Generally, when some or all of the brokers are competing for the same niche, a gap is created in the coverage of categories, adversely affecting some consumers.