:: QUESTION ::
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" Hey, I’m trying to solve a game using Bayes-Nash equilibrium. The two players are a prisoner who would stab or don't stab and the second player is a prison guard who will tase or don't tase. Either way the guard best option will always be to tase. How can I find a Bayes-Nash equilibrium for this problem? "
NOTE ::  If you like would like to  review - Bayesian Nash Equilibrium check out this video
  • PART ONE :: ASSUMPTIONS
To answer this question, we first have to make some assumptions about the scenario given to us. Remember that for a Bayesian game, there is one additional player that exists in the situation - Nature. Nature is a non-strategic player that is represented in the decision node tree to endow players with private information about their payoffs in certain circumstances.

In this case, we know that the options for the Prison Guard are to “Tase” and “Not Tase”, while that of the Prisoner is to “Stab” or “Don’t Stab”. While the Prison Guard’s best option is always to tase, we are unsure of what the prisoner’s best choice is; this would be dependent on the prisoner’s nature. If they are violent, the prisoner’s payoffs would be different than if they are peaceful, due to the presence or absence of guilt for hurting someone who was unarmed. 

  • PART TWO :: SETTING UP PAYOFFS 
Therefore, we can represent this game’s payoffs in the following payoff matrices.

Note: the exact number assigned to these payoffs do not necessarily matter as long as they fit the conditions of the question presented. 

When filling in the payoffs for the prisoner’s violent nature, we can assume that stabbing an undefended guard would give them a higher payoff than if the guard had also tased them. Likewise, not choosing to stab when they are tased would only lead to a negative payoff. If neither attack, then there is no positive or negative outcome.

When filling the payoffs for the prisoner’s peaceful nature, we can assume that stabbing an undefended guard would give them a negative payoff, since they would not want to attack someone who is undefended. On the other hand, the prisoner choosing not to stab when they are tased would still produce a negative payoff, since they are still hurt when sticking with their pacifist beliefs. Therefore, the only time the peaceful prisoner would have a positive payoff is if they chose to not stab and the prison guard also chose not to tase. If the prisoner stabbed when the prison guard tased, the payoff for the prisoner would be neither positive nor negative, since they broke from their pacifist rules but still was vindicated through self-defense by the prison guard’s tase. We can see that regardless of the prisoner’s nature, the payoffs for the prison guard is always highest when they decide to tase as fitting with the question’s conditions.
  • PART THREE :: DETERMINING BEST STRATEGY
Converting the payoff matrix of the players into the following extensive form/game tree, we can see that the nodes circled represent the information set for each player, which helps to showcase that the prison guard is unsure of which game they are playing, since they do not know the prisoner’s nature. Remember that the information set represents all the possible moves that could have taken place in the game so far, given what the player has already observed. The probability of each nature occurring is assumed to be 50% as there were no other probabilities given in the question, and we can presume that each nature is equally probable. Therefore, to solve for the Bayesian Nash equilibrium of this game, we have to test the strategy profiles of each player to see if they are the best responses for this player. Since we know that tasing (represented as “T”) is always the best option for the prison guard, the strategy profiles we will be testing are (T, (S,NS)) and (T, (NS,S)), where “S” represents the action stab, “NS” represents no stabbing, and the order of the letters is indicative of the violent and peaceful type, respectively. To check for best responses, we can calculate each player’s expected payoffs based on these strategy profiles. 
  • PART FOUR :: CALCULATING EXPECTED PAYOFFS
Note :: Since we are only looking at the payoffs for the guard when they tase, we can simplify our game tree to represent only those decision possibilities.
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Starting with the guard’s payoffs for (T, (S,NS)), we look at the extensive form of the game and follow the decision nodes to calculate the expected payoffs, taking into account the probability of such choices occurring based on the nature of the prisoner. Then we can calculate the payoffs for the prisoner in the (T, (S,NS)) profile, with the expected payoffs being 3/2 for the guard and 0 for the prisoner. Repeating this process for the (T,(NS,S)) profile, we see that the guard’s expected payoffs are again 3/2 but the prisoner’s is now -1.
We could double check that any strategy profile that includes not tasing would not be favorable through the following calculations. We see that the payoffs in both strategy profiles is -1/2 for the guard, which is lower than the previous 3/2 payoffs. As for the prisoner's payoff, we again see that they receive a higher payoff when choosing stab for their violent nature and no stab for their peaceful nature (3/2 versus -1/2). 

​ANSWER: Therefore, the Bayesian-Nash equilibrium is (T, (S,NS)). Hope this helped!

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