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The JIT compiler (optimizer) counts how many times each piece of code is executed and compiles them after a threshold is achieved (1500 executions in client mode, 10000 in server mode). Compilation is done in a background thread, and that piece of code stays in interpreted mode until the background compilation finishes. So, the optimization runs in background and should not interfere unless you limit the JVM to a single core. In client mode, optimization finishes faster and the JVM consumes less memory, but the resulting compiled code is slower. Server mode is the opposite.

You can change the threshold with the -XX:CompileThreshold=10000 option, change the optimizer to run in foreground with the -Xbatch option, choose client mode with the -client option, and server mode with the -server option. Experimenting with different settings may help understand what is going on.

Maybe it is skipping turns while it is still running in interpreted mode (beginning of first round), after the bot history grows a bit. After the code switches to compiled mode, it becomes faster and stops skipping turns.

Warming the JVM with pointless calculations probably won´t work since the JIT compiler will compile only that piece of pointless code. And sometimes, warming up confuses the optimizer and it activates the wrong set of optimizations, resulting in slower compiled code.