Our baseline (once again):
Agent 1 (nash distribution): .296
Agent 2 (nash distribution): .296
Agent 3 (nash distribution): .296
Another case:
Agent 1 (always chooses 1): .489
Agent 2 (even distribution -- equal chance of any number 1-10 being chosen): .411
Agent 3 (nash distribution): .054
In this situation, the nash strategy comes out far far behind either of the other two strategies.
This makes sense intuitively:
Agent 3 chooses 1 nearly half (45.6%) of the time. It will lose with that choice every time because Agent 1 chooses 1 every time.
When Agent 3 chooses 2-10 (100 - 45.6 = ) 54.4% of the time, it will lose almost every time because Agent 1 already chose 1.
The only case where Agent 3 wins is when Agent 3 chooses a value 2-10 (54.4%) AND Agent 2 chooses 1 (10%), eliminating itself.
54.4% * 10% = 5.4%, which is exactly the value discovered above.
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The specific strategies chosen by your competitors have a very large impact on your strategy's effectiveness.
I fail to understand how this can be considered an optimal strategy.