I present and analyze the patterns for expected profits of the agents for the discrete model in the case when a bad signal arrives in each period. Figures 9.1 and 9.2 plot the expected profits of agents 1 and 2, respectively, as they change over time.
Each of the curves on the expected profits charts has exactly one peak. This peak occurs at different times depending on the agent and the case. For agent 1, the cases for large high threshold (blue and green curves) offer high expected profits in early periods. Expected PnL then continues to rise, more so for the blue curve – since in dates 4 and 5 agent 1 is bullish
and of type 2, while agent 2 already has type 3. So there is a large difference of opinions, while agent 2 is less likely to adjust his opinion to potential high signals, leading to greater expected profits for agent 1, than if agent 2 was of type 2. After that expected profits start to drop as the agents start to converge in their beliefs. The drop is more rapid since agent 1 becomes type 3 faster due to the higher low threshold at which point expected profits are very low.
Continuing with agent 1, the cases for small high threshold (red and purple curves) offer lower profits in early periods. The profits then continue to rise; more rapidly for the red curve – again because agent 2 becomes type 3 faster. For the purple curve, profits first rise slowly, and then grow very rapidly, as agent 1 still has type 1, but agent 3 has type 3, so there is a large difference in opinions. As with the blue and green curves, the expected profits then drop as the beliefs converge, and agent 1 becomes type 3 along with agent 2 who by that time has switched to type 3 several periods earlier.
The pattern for agent 2 is similar. It is interesting that the curves with similar high (low) threshold are a lot more similar for agent 2 than for agent 1. This is likely because the peak occurs for agent 2 when he has type 2 (or has just become type 3) – since after that he pushes the price too much and is too convinced in his beliefs to expect a large difference in opinions later on. Thus the location of the high threshold matters a lot more for agent 2.
0.00
Agent 1 Expected Profits vs Time
Th1 = 80%,
Figure 9.1: Expected profits of agent 1 as a function of time.
0
Agent 2 Expected Profits vs Time
Th1 = 80%,
Figure 9.2: Expected profits of agent 2 as a function of time.
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