The Pregnancy Test Standoff

In the movie Kill Bill, a female assassin targeted the main character Beatrice Kiddo. Just a few minutes before the assassin was set to deliver her bunch of flowers to Beatrice’s hotel room, Beatrice learned that she was pregnant.

The assassin then knocked on the door disguising herself as a hotel worker carrying a bunch of flowers. Beatrice then checked to see who was on the door but her pregnancy kit fell. And just at the right time that Beatrice bent her body to pick up the kit, the assassin pulled the trigger making a big hole on the door and barely missing Beatrice’s head.

And what follows is the pregnancy test standoff. Beatrice is holding a handgun aimed at the assassin’s head and the assassin aiming her big gun at Beatrice. Then a little talk ensued making them to exchange valuable information that they both realized were important in their lives.

Beatrice then told the assassin that she is the deadliest assassin in the world, something that the assassin knew to be an exaggeration of Beatrice’s talent but that was enough to make her know what Beatrice is capable of. Beatrice then said that she is pregnant as if to let her know that she is more deadly at that time as she is protecting her child, and that there’s not much payoff for the assassin to kill a woman and the innocent baby inside. It is up to the assassin to believe what Beatrice said.

So we model the situation with nature moving first choosing the type of Beatice – a pregnant and deadly Beatrice or just a regular woman who just happened to have a gun – and the Chinese assassin forms her beliefs. Beatrice moves first. Her move is to ask the assassin to back away or to agree to fight. The assassin then fights or not. The assassin does not know the type of Beatrice and so makes belief. Because Beatrice was able to dodge the initial shot, she believes what Beatrice said with probability 80%. The diagram below depicts the payoff.

When there is not fight, both walk free and get 20. If the assassin is asked not to fight and she fights a deadly Beatrice she will lose fighting a more careful fight and gets 0. If she just fought a regular Beatrice she will win and gets 10.

If the deadly Beatrice chooses to fight without explaining who she is, the assassin will lose badly and gets -10. If the regular Beatrice just fights, the assassin will win and Beatrice will lose badly.

Perfect Bayesian Nash Equilibrium

We check the separating equilibrium where the deadly Beatrice fights and the regular Beatrice asks her opponent not to fight.

{FA,FF’} => EU2 = -10*.8 + 10 * .2 = -6
{FA,FN’} => EU2 = -10*.8 + 20 * .2 = -4
{FA,NF’} => EU2 = 0*.8 + 10 * .2 = 2
{FA,NN’} => EU2 = 0*.8 + 20 * .2 = -6

We then check {FA,NF’} for equilibrium. The deadly Beatrice would not want to deviate because her payoff of 20 is better than 10. The regular Beatrice would want to deviate because her new payoff of 20 is better than her current payoff of 0. Thus {FA,NF’} is not PBE.

We next check the separating equilibrium where the deadly Beatrice asks her opponent not to fight and the regular Beatrice fights.

{AF,FF’} => EU2 = 0*.8 + 10 * .2 = 2
{AF,FN’} => EU2 = 20*.8 + 10 * .2 = 18
{AF,NF’} => EU2 = 0*.8 + 0 * .2 = 0
{AF,NN’} => EU2 = 20*.8 + 0 * .2 = 16

We then check {AF,FN’} for equilibrium. The deadly Beatrice would not want to deviate because her payoff of 20 is better than 10. The regular Beatrice would not want to deviate because her current payoff of 20 is better than the payoff of 0 if she deviates. Thus {AF,FN’} is PBE.

We then check for pooling equilibrium when both types of Beatrice fight. {FF,FF’} => EU2 = -10*.8 + 10 * .2 = -6
{FF,FN’} => EU2 = -10*.8 + 10 * .2 =