Idealistic
Junior Member
- Joined
- Sep 7, 2007
- Messages
- 97
In the county of Utopia the voters are evenly distributed along the political spectrum from left (a = 0) to right (a = 1). There are two candidates (Oddette and Steven) for Dog Catcher. Each voter casts his vote for the candidate closer to his political position. Oddette and Steven get to declare their political position (i.e choosing values o and s respectively where 0 ? o ? 1 and 0 ? s? 1). If o = s then they split the vote. If both get the same number of votes then the decision is made by tossing a coin.
a) What are the payoff functions, i.e. number of votes? (Note: this is not a matrix nor a game tree problem!)
b) Show that (o,s)=(1/2,1/2) is the only Nash Equilibrium.
Heres what I have:
For o = s --> (1/2, 1/2)
If o doesnt equal s, then
Number of Votes for Odette:
(o + s)/2 for o > s and (1 - (o + s)/2) for s > o
Number of votes for Steven:
(1 - (o + s)/2) for o < s and (o + s)/2 for s > o
Im not sure if this is entirely correct however. It would be nice if I could combine these into two separate formulas (one for Oddete and one for Steven). Im not really sure on how to start b).
a) What are the payoff functions, i.e. number of votes? (Note: this is not a matrix nor a game tree problem!)
b) Show that (o,s)=(1/2,1/2) is the only Nash Equilibrium.
Heres what I have:
For o = s --> (1/2, 1/2)
If o doesnt equal s, then
Number of Votes for Odette:
(o + s)/2 for o > s and (1 - (o + s)/2) for s > o
Number of votes for Steven:
(1 - (o + s)/2) for o < s and (o + s)/2 for s > o
Im not sure if this is entirely correct however. It would be nice if I could combine these into two separate formulas (one for Oddete and one for Steven). Im not really sure on how to start b).