Battle of the Couples
Bruce: I have waited for this game since the start of the season. We will be hosting our arch rivals in the final game of the season. And like that is not enough excitement for me, if we win we will crown as champions: It is either us or them.
Every year when I decide to buy a season ticket, I dream of going to this game. I haven’t missed a game since 2006 and this season’s game will be cherry of top in case we win. We have to win and I must witness it.
Jane: My favorite band is retiring and they are on the road for the last time. Luckily for me they will be visiting my town this time: I will have the opportunity to watch them alive in my city for the first and the last time.
When I was 11, I bought a random album. I went home, pressed start on my music box and fell in love with them instantly. It was their best-selling album from 1992. I have been laughing with their songs, I have been crying with their songs. I had spent my youth with them. And I will say farewell to them in person.
Bruce and Jane have been together for some time, but they are in a pickle now. Derby game is postponed and both events will be started at the same day, same hour.
In this scenario we assume that Jane and Bruce are equally (or at least equivalently) in love with each other.
Since they are ideal couple, spending time together is the most important thing for them. That is why we’ll give them 1 point for being together if they choose to go to the same event.
We’ll give them 1 extra point for their favorite event: For Jane concert is 1 extra point and for Bruce game is 1 extra point.
If they are not together, they get -1 point as they are unhappy for being apart. This is why in case they go to separate events they will get 0 point each.
Then we can construct the game matrix as follows:
In this game if players choose to be selfish and only consider their happiness, Jane would choose to go to the concert as Bruce would choose to go to the game. We already know that the outcome of concert-game is (0,0). In order to test if this is the Nash equilibrium or not, we must test players one by one. First let Bruce chooses first. He will decide to go to the game.
If Jane knows about it, she will have to choose going to the game also since outcome of the game (1) is larger than the outcome of the concert (0) for her. In this case the equilibrium is game-game.
Now let Jane do the first selection. She will choose to go to the concert:
In this case Bruce has to choose to go to the concert as its outcome (1) is larger than game’s outcome (0). Here, equilibrium is concert-concert.
We just found out that there are two Nash equilibriums in this game: Concert-concert or game-game. Both cases could happen if only Jane and Bruce are willing to cooperate. Otherwise if they act selfish, they will get no happiness whatsoever.
Personal Thought: Even with an ideal couple Bruce should compromise since “she” will always win.
Jane Doesn’t Love Bruce
Jane thinks she could do better. And she has a point: Bruce is a 43-year old unsuccessful computer engineer who is slowly going bald. On the other hand Bruce is very much so in love with Jane and deep inside he knows that she is a catch.
For Jane, being in the concert means more.
For Bruce, being with Jane and going to the game are equally important.
Let’s build our game with this information.
If they both decide to go to the concert: Jane will receive 2 points (0 from being with Bruce, 2 from being at the concert). Bruce will get 1 point and that comes from being with Jane.
If they both decide to go to the game: Jane will receive -1 point (0 from being with Bruce, and -1 from not being in the concert). Bruce will get 2 points: 1 from being with Jane, the other from being at the game.
Jane to the concert, Bruce to the game: Jane will get 2 points in total, all coming from being at the concert. Bruce will get 1 point in total and that comes from being at the game.
Jane to the game, Bruce to the concert: Jane will be furious and get -1 points as Bruce will get 0 point.
Now we can construct the matrix of the game as follows:
If both players are selfish, Jane would choose to go to the concert as Bruce would choose to go to the game. In this scenario concert-game becomes the result of the game. Its outcome is (2,1).
We should check if concert-game is the Nash equilibrium for this game.
If Jane knows that Bruce is going to the game, she would have two options: 2 and -1. Obviously she would choose 2; that is going to the concert. Then in this game result becomes concert-game.
If Bruce knows that Jane is going to the concert, his choices would get him 1 point in either case. This is why Bruce would have two identical choices. Concert-concert and concert-game will have the same probabilities.
In conclusion, there are two Nash equilibriums for this game: Concert-concert and concert-game. Both cases have the same outcome that is (2,1).
Find the matrix of the game and its Nash equilibrium when Bruce doesn’t love Jane while Jane does love him.