Game Theory

2. Consider three mobile cantaloupe vendors (vendor A, vendor B, and vendor C) that go to sell perfect substitute goods at one of two locations on a street each day. One location is right in the middle of the street (at o). The other is 3K4 the way down the street (halfway between 0 and one end). Consumers are located uniformly on the street and will always buy from the closest vendor. The vendors all sell for the same price. If one vendor gets stuck between the two other vendors at the same location, then that vendor will not sell any product, as they will be cutoff from both sides. Find the three Pure Strategy Nash Equilibria. Find a Mixed Strategy Nash Equilibrium. Clues: - label locations 0 list all the possible outcomes and make up some payoffs associated with them, e.g., if all 3 vendors choose to go to 0, then two vendors will get half the demand and one will get nothing, the one who gets nothing is random, p = U3 {this is not an MSNE probability} a you will have to set up a system of equations with three equations and three unknowns, maybe use Wolfram Alpha {google it} to solve these equations 0 draw out the game with a combination of a game tree and normal forms: the rst vendor chooses a location (2 branches) and each branch leads to a normal form, there are three payoffs associated with each outcome I breathe Street : 0= center of street R = 14 way down street Example of an outcome : Plat O ( left side ) P2 at O ( right side ) p3 at R The liegram shows regions of demand given this outcome :Pl gets 50% P2 get 14 of 50% = 12.5% P3 gets 3/4 of 50% = 37.5% Note : these are payoffs AFTER " nature " . Why did PV get 50% ! 12 get 12.5% ? - Nature chooses , so you need to do expected value : E ( PI) - 2 ( 50 ) + E ( 12.5) = 31.25% E ( P2 ) = 2 ( 50) + E ( 12.5) = 31.25%% So if more than are player shows up at a location , you need to calculate expected values . Then you can draw gamefree / normal Farms ; PI 2 O P3 - - R 2 R R P2 P2 O 31. 25, 31.25 37.5 Simultaneous Zoom in move game 31.25 , 31.25, 37.5 each outcome has 3 lexpected ) payotts .Once you have all outcomes you can set up 3 equations wl 3 unknowns make players indifferent between O and R