In the original problem you presented (the one without any further crimes), I don't think testifying is always the best option. As Alice, if you are both aware of your mutual rationality and know nothing about each other but that, wouldn't a better option be biased randomization? Consider the probability of Bob testifying against you to be 50% and calculate the optimal probabilities for your decision in order to minimize the number of years in prison (negative utility). You get a 7/13 chance of testifying against him. However, remember the other player is rational and that the game is symmetric, so he will also have the get the same probability as you. But, that being the case, you can no longer think his strategy will be to just "flip a coin". Naturally, you compute again, and again, ad infinitum. What you will find that the value quickly converges towards approximately 58.01% chance (more accurately, 58.01119204342238...%) of testifying against Bob (the same thing goes for him). This way, the overall expected utility is roughly -10.0958952, and no other strategy in the given setting (that I'm aware of) will lead to a greater value. If I made some mistake in my reasoning, please clarify. I am a little inexperienced in the subject and only started learning about game theory today.
In the original problem you presented (the one without any further crimes), I don't think testifying is always the best option. As Alice, if you are both aware of your mutual rationality and know nothing about each other but that, wouldn't a better option be biased randomization? Consider the probability of Bob testifying against you to be 50% and calculate the optimal probabilities for your decision in order to minimize the number of years in prison (negative utility). You get a 7/13 chance of testifying against him. However, remember the other player is rational and that the game is symmetric, so he will also have the get the same probability as you. But, that being the case, you can no longer think his strategy will be to just "flip a coin". Naturally, you compute again, and again, ad infinitum. What you will find that the value quickly converges towards approximately 58.01% chance (more accurately, 58.01119204342238...%) of testifying against Bob (the same thing goes for him). This way, the overall expected utility is roughly -10.0958952, and no other strategy in the given setting (that I'm aware of) will lead to a greater value. If I made some mistake in my reasoning, please clarify. I am a little inexperienced in the subject and only started learning about game theory today.