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M4.4 Mixed Strategy Games

When there is no saddle point, players will play each strategy for a certain percentage of the time. This is called a mixed strategy game. The most common way to solve a mixed strategy game is to use the expected gain or loss approach. The goal of this approach is for a player to play each strategy a particular percentage of the time so that the expected value of the game does not depend on what the opponent does. This will occur only if the expected value of each strategy is the same.

Consider the game shown in Table M4.4. There is no saddle point, so this will be a mixed strategy game. Player Y must determine the percentage of the time to play strategy $Y_{1}$ $Y_{1}$ and the percentage of the time to play strategy $Y_{2} .$ $Y_{2} .$ Let P be the percentage of time that player Y chooses strategy $Y_{1}$ $Y_{1}$ and $1 - P$ $1 - P$ be the percentage of time that player Y chooses strategy $Y_{2}$ $Y_{2}$ . We must weight the payoffs by these percentages to compute the expected gain for each of the different strategies that player X may choose.

Table M4.4 Game Table for Mixed Strategy Game

	PLAYER Y’s STRATEGIES
		$Y_{1}$ $Y_{1}$	$Y_{2}$ $Y_{2}$
PLAYER X’s STRATEGIES	$X_{1}$ $X_{1}$	4	2
	$X_{2}$ $X_{2}$	1	10

Table M4.4 Full Alternative Text

For example, if player X chooses strategy $X_{1},$ $X_{1},$ then P percent of the time the payoff for Y will be 4, and $1 - P$ $1 - P$ percent of the time the payoff will be 2, as shown in Table M4.5. Similarly, if player X chooses strategy $X_{2}$ $X_{2}$ , then P percent of the time the payoff for Y will be 1, and $1 - P$ $1 - P$ percent of the time the payoff will be 10. If these expected values are the same, then the expected value for player Y will not depend on the strategy chosen by X. Therefore, to solve this, we set these two expected values equal, as follows:

4 P + 2 (1 - P) = 1 P + 10 (1 - P)

$4 P + 2 (1 - P) = 1 P + 10 (1 - P)$

In Action Game Theory in the Brewing Business

Companies that understand the principles and importance of game theory can often select the best competitive strategies. Companies that don’t can face financial loss or even bankruptcy. The successful and unsuccessful selection of competitive gaming strategies can be seen in most industries, including the brewing industry.

In the 1970s, Schlitz was the second-largest brewer in the United States. With its slogan “the beer that made Milwaukee famous,”* Schlitz was chasing after the leader in beer sales, Anheuser-Busch, maker of Budweiser. Schlitz could either keep its current production output or attempt to produce more beer to compete with Anheuser-Busch. It decided to get more beer to the market in a shorter amount of time. In order to accomplish this, Schlitz selected a strategy of distributing “immature” beer. The result was cloudy beer that often contained a slimy suspension. The beer and Schlitz’s market share and profitability went down the drain. Anheuser-Busch, Miller, and Coors became the market leaders.

Similarly, when Miller first decided to market Miller Lite, with the slogan “tastes great—less filling,” Anheuser-Busch had two possible gaming strategies: to develop its own low-calorie beer or to criticize Miller in its advertising for producing a watered-down beer. The strategy it selected was to criticize Miller in its advertising. The strategy didn’t work, Miller gained significant market share, and Anheuser-Busch was forced to come out with its own low-calorie beer—Bud Light.

Today, Anheuser-Busch, Miller Coors, Pabst Brewing Company (owner of the Schlitz brand), D. G. Yuengling and Son Company, Boston Beer Company, and other large beer manufacturers face new games and new competitors. Although it is too early to tell what the large beer makers will do and how successful their strategies will be, it appears that their strategy will be to duplicate what these new competitors are doing. What is clear, however, is that a knowledge of the fundamentals of game theory can make a big difference.

*From “American Brewing, Unreal” by Philip Van Munching, © September 4, 1997, The Economist Newspaper Limited.

Source: Based on Philip Van Munching, “American Brewing, Unreal,” The Economist (September 4, 1997): 24, © Trevor S. Hale.

Solving this for P, we have

P =^{8} /_{11}

$P =^{8} /_{11}$

and

1 - P = 1 -^{8} /_{11} =^{3} /_{11}

$1 - P = 1 -^{8} /_{11} =^{3} /_{11}$

Thus, $^{8} /_{11}$ $^{8} /_{11}$ and $^{3} /_{11}$ $^{3} /_{11}$ indicate how often player Y will choose strategies $Y_{1}$ $Y_{1}$ and $Y_{2}$ $Y_{2}$ , respectively. The expected value computed with these percentages is

1 P + 10 (1 - P) = 1 (^{8} /_{11}) + 10 (^{3} /_{11}) =^{38} /_{11} = 3.46

$1 P + 10 (1 - P) = 1 (^{8} /_{11}) + 10 (^{3} /_{11}) =^{38} /_{11} = 3.46$

Performing a similar analysis for player X, we let Q be the percentage of the time that strategy $X_{1}$ $X_{1}$ is played and $1 - Q$ $1 - Q$ be the percentage of the time that strategy $X_{2}$ $X_{2}$ is played. Using these, we compute the expected gain shown in Table M4.5. We set these to be equal, as follows:

4 Q + 1 (1 - Q) = 2 Q + 10 (1 - Q)

$4 Q + 1 (1 - Q) = 2 Q + 10 (1 - Q)$

Table M4.5 Game Table for Mixed Strategy Game with Percentages (P, Q) Shown

		$Y_{1}$ $Y_{1}$	$Y_{2}$ $Y_{2}$
		P	$1 - P$ $1 - P$	Expected gain
$X_{1}$ $X_{1}$	Q	4	2	$4 P + 2 (1 − P)$ $4 P + 2 (1 − P)$
$X_{2}$ $X_{2}$	$1 - Q$ $1 - Q$	1	10	$1 P + 10 (1 − P)$ $1 P + 10 (1 − P)$
Expected gain		$4 Q + 1 (1 − Q)$ $4 Q + 1 (1 − Q)$	$2 Q + 10 (1 − Q)$ $2 Q + 10 (1 − Q)$

Table M4.5 Full Alternative Text

In Action Using Game Theory to Shape Strategy at General Motors

Game theory often assumes that one player or company must lose for another to win. In the auto industry, car companies typically compete by offering rebates and price cuts. This allows one company to gain market share at the expense of other car companies. Although this win–lose strategy works in the short term, competitors quickly follow the same strategy. The result is lower margins and profitability. Indeed, many customers wait until a rebate or price cut is offered before buying a new car. The short-term win–lose strategy turns into a long-term lose–lose result.

By changing the game itself, it is possible to find strategies that can benefit all competitors. This was the case when General Motors (GM) developed a new credit card that allowed people to apply 5% of their purchases to a new GM vehicle, up to $500 per year, with a maximum of $3,500. The credit card program replaced other incentive programs offered by GM. Changing the game helped bring profitability back to GM. In addition, it helped other car manufacturers who no longer had to compete on price cuts and rebates. In this case, the new game resulted in a win–win situation with GM. Prices, margins, and profitability increased for GM and for some of its competitors.

Source: Based on Adam Brandenburger, et al., “The Right Game: Use Game Theory to Shape Strategy,” Harvard Business Review (July–August 1995): 57, © Trevor S. Hale.

Solving for Q, we get

Q =^{9} /_{11}

$Q =^{9} /_{11}$

and

1 - Q = \frac{2}{11}

$1 - Q = \frac{2}{11}$

Thus, $^{9} /_{11}$ $^{9} /_{11}$ and $^{2} /_{11}$ $^{2} /_{11}$ indicate how often player X will choose strategies $X_{1}$ $X_{1}$ and $X_{2}$ $X_{2}$ , respectively. The expected gains with these probabilities will also be $^{38} /_{11},$ $^{38} /_{11},$ or 3.46.

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Table of Contents for M4.4 Mixed Strategy Games

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Table of Contents for
M4.4 Mixed Strategy Games