I started reading up on some Game Theory and started analysing the Battle of the Sexes game, using wikipedia as source (link). In the explanation they link to an article (link, reading section 4.5) that says that letting one person have the option of 'Burning money' and lowering their payout across the board can force rational players to go to the favored stategy of one of the players.
To me, the reasoning seems incorrect. To my understanding the BoS game is not sequential game. If it would have been sequential, a rational player would always choose their favorite choise causing the second rational player then always chosing that option as well.
Now if one player has the option of burning money, it is state that the second player observes the fact that player one plays 'Burn' or 'Do not burn' and then both players make their choise (to my understanding: simultaneously). If player oen plays 'Burn', the matrix has less payout for player one while player two receives the same payout. It's still the same Battle of the Sexes problem, but now player one just has less payout but still has the same optimal payout choises.
The linked article instead has a payout matrix that introduces 4 strategies for player one:
- 'Burn', then choose own favored option.
- 'Burn', then choose other player's favored option.
- 'Do not burn', then choose own favored option.
- 'Do not burn', then choose other player's favored option.
The listed strategies, according to the article, for player two: 1. Choose player one's favored option. 2. Choose own favored option. 3. Choose player one's favored option, unless player one burns. In that case, chose own favored option instead. 4. Choose own favored option, unless player one burns. In that case, chose player one's favored option instead.
I can't agree with the article that player two has strategies 3 and 4 available. Player two only has to observe the 2x2 payout matrix that exists after player one has made their choise to burn or not.
One could argue that introducing 'Burn' is similar to signaling your choise, like saying 'I'm going to play rock' in Rock-Paper-Scissors. In the Rock-Paper_scissors game, it's optimal to ignore such a signal. I would argue similarly to for player one playing or not playing the 'Burn' option. If could signal 'I'm lowering my highest possible payout to shown to make you favored choise more valuable compared to my favorable choise', but could also mean 'I want my option so badly, I'm even torching my gain from it to make it clear how much I want it'. Or perhaps some just want to world to burn, who knows. As such, I cannot see how player two 3 and 4 stategies are valid options at all.
Am I missing something here?