The Minimax Strategy

A new coca - cola company, Scumcola, entered the market.There are three possibilities for advertising. Their major competitor, Crapcola, also has three choices of advertising. Previous experience implies the following payoff matrix.
Scumcola\Crapcola 1 2 3
1 20,000 30,000 70,000
2 10,000 40,000 60,000
3 90,000 50,000 80,000
Each company wants to find their optimal strategy. Scumcola wants to attract the maximum number of customers from Crapcola, and Crapcola wishes to minimise the loss of customers to Scumcola.
We use the minimax procedure. Scumcola realises that Crapcola wishes to minimise the loss of custom, so they consider the minimum loss of custom to Crapcola for each choice of campaign. Scumcola sees that the minimum gain for campaign 1 is 20,000 customers, the minimum gain for campaign 2 is 10,000 customers and the minimum gain for campaign 3 is 50,000 customers. Thus Scumcola will choose strategy 3, which mas the maximum of the row minimums.
Crapcola realises that Scumcola wants to choose the maximum for each of Crapcola's strategies, so Crap cola only considers the column maximums. For strategy 1 the maximum loss is 90,000 customers, for strategy 2 the maximum loss is 50,000 customers and for stratify 3 the maximum loss is 80,000 customers. Thus Crapcola will choose strategy 2, which has the minimum of the column maximums. In this way, both companies will minimise their losses. The entry in the payoff matrix corresponding to both players' choices is called the saddle point. Neither company will ever change their strategy since their chosen strategy is the best for them. This type of game is called a 'two player zero sum game', because each players losses become the other players gain.
Alternatively, Scumcola sees that column 3 contains the largest number in each column, representing the largest payoff to the company, so 3 is the best strategy regardless of which strategy Crapcola chooses. Crapcola notices this too, and since Scumcola is sure to choose this strategy, Crapcola will choose strategy 2 which minimises their loss in this case.

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