We did not complete a powerpoint, but we briefly explained the concept of game theory and gave the following twoexamples:
[1] Frontier and United both plan to add another city to their list of cities with nonstop flights from Denver. Milwaukee, Albany, Hartford are the three choices. If both airlines add Milwaukee to their list then Frontier will show a gain of $75,000 over United; if both add Albany, Frontier will show a gain of $50,000 over United. However if both choose Hartford, then United shows a gain of $100,000 over Frontier. If Frontier chooses Albany and United chooses either of the other two, then Frontier shows a profit of $200,000 over United. If Frontier chooses Milwaukee and United chooses Hartford or vice versa, then they are even. Finally, if United chooses Albany and Frontier chooses Milwaukee, then Frontier has a gain of $25,000 over United, but if Frontier chooses Hartford, United has a gain of $50,000 over Frontier. Give the payoff matrix and equilibrium point.
UNITED
Milwaukee
Albany
Hartford
Milwaukee
$75,000
$25,000
$0
FRONTIER
Albany
$200,000
$50,000
$200,000
Hartford
$0
-$50,000
-$100,000
Using the dominant strategy to find the equilibrium point, you can cross out row 1 and row 3 because the row player would like the highest numbers because everything is listed in their terms of what they will gain. The second row dominates row 1 and row 3 because the values are higher. With the row 2 being the only one remaining, you take the viewing of the column player. The column player would like the lowest number, so that would mean they would be losing less since the game is constructed in terms of the row player.
UNITED
Milwaukee
Albany
Hartford
Milwaukee
$75,000
$25,000
$0
FRONTIER
Albany
$200,000
$50,000
$200,000
Hartford
$0
-$50,000
-$100,000
Therefore the equilibrium point for the game would be Albany. Frontier will make a profit of $50,000 and United will only lose $50,000.
[2] Home Depot and Lowes are planning to locate stores in eastern Colorado. If Home Depot locates in Sterling and Lowes locates in Limon, then Home Depot can expect an annual profit of $150,000 more than Lowes. If both locate in Sterling they can expect an equal profit. If Home Depot locates in Limon and Lowes locates in Sterling, Lowes can expect an annual profit of $125,000 more than Home Depot. If they both locate in Limon, then Lowes profit exceeds Home Depot's by $25,000. Write the payoff matrix of the game and the equilibrium.
LOWES
Sterling
Limon
Sterling
$0
$150,000
HOME DEPOT
Limon
-$125,000
-$25,000
In order to find the equilibrium point of this game, you would use the min-max strategy. For each strategy of the row player (Home Depot), find the strategy of the column player (Lowes) corresponding to the least element, which is the lowest payoff for Home Depot. Since this is a zero-sum game, this is the highest payoff for Lowes. The least element of row 1 is 0 (Sterling, Sterling), and the least element of row 2 is -125,000 (Limon, Sterling). Since Home Depot wants the highest profit, Home Depot would choose to locate in Sterling; so the max-min is 0 (Sterling, Sterling). Then, for each strategy of the column player (Lowes), find the strategy of the row player (Home Depot) corresponding to the greatest element, which is the highest payoff for Home Depot. The greatest element of column 1 is 0 (Sterling, Sterling), and the greatest element of column 2 is 150,000 (Sterling, Limon). Since Lowes wants the highest profit, which is the lowest profit for Home Depot, Lowes would choose to locate in Sterling; so the min-max is 0 (Sterling, Sterling). Since (Sterling, Sterling) is simultaneously the max-min and min-max of the game, the equilibrium point is both Home Depot and Lowes locating their stores in Sterling. The value of the game is 0; the two companies would break even if both were to build their stores in Sterling.
