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Determining the Ideal Amount of Advertising
Chapter 12: Game Theory: Fun Only if You Win
business situations, if you move first, your rival may be able to neutralize your decision by moving second. You need to develop a decision-making rule that helps you anticipate your rival’s decision.
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Decision-makers use backward induction in sequential games. Backward induction means that you develop your decision by looking at the future, or at how the game ends. Consider the decision tree in Figure 12-4. Two car dealers, Cathy’s Cars and Otto’s Autos, must decide whether to raise, lower, or charge the same price on their used cars. Because of her leadership position in the market, Cathy’s Cars chooses her used car price first, and then Otto’s Autos follows. When establishing her price, Cathy needs to consider how Otto responds by using backward induction. Using backward induction requires the following steps. Decision-makers start at the end of the game. The player making the last decision chooses the decision that has the greatest payoff. As a result, the player making a decision in the step before the last decision takes into account the likely decision in the last step when making her decision. Decision-makers continue working backward from the last decision, taking into account the likely decision made at each step in the sequence, until the first step — the initial decision step — is reached. Figure 12-4 illustrates the backward induction process.
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Figure 12-4: Backward induction.
Figure 12-4 summarizes the various outcomes that are possible in the situation. The far left starts with Cathy’s possible choices — she can raise price, charge the same price, or lower price. In the middle, Otto’s choices are portrayed for each possible choice Cathy can make. At the far right are the annual profits each dealer receives given their combined choices. For example, if Cathy lowers price, follow the lower branch from Cathy’s Cars to Otto’s Autos. If Otto charges the same price, follow the middle branch to the outcome — $92,000 profit for Cathy and $68,000 profit for Otto. Thus, when making her decision, Cathy needs to consider how Otto responds.
1. If Cathy raises price, Otto charges the same price to earn $94,000.
Otto charges the same price because $94,000 is a higher profit for Otto than $84,000 if he raises price or $73,000 if he lowers price. As a result,
Cathy’s profit is $66,000.
2. If Cathy charges the same price, Otto lowers price to earn $87,000.
Otto lowers price because $87,000 is a higher profit than $79,000 if
Otto raises price or $76,000 if he charges the same price. Cathy’s profit is $71,000 — the result of her charging the same price and Otto lowering price.
3. If Cathy lowers price, Otto charges the same price to earn $68,000.
Otto charges the same price because $68,000 is a higher profit than $25,000 if Otto raises price or $63,000 if he lowers price. Cathy’s profit is $92,000.
4. Cathy’s decision is to lower price.
When Cathy lowers her price, Otto charges the same price, resulting in $92,000 profit for Cathy and $68,000 profit for Otto. The $92,000 profit for Cathy is the best she can expect after Otto makes his choice. As indicated in Step 1, if Cathy raises price she ultimately receives $66,000 in profit and if she charges the same price, she receives $71,000 in profit as indicated in Step 2.
By using backward induction to anticipate Otto’s response to her decision, Cathy is able to choose the best possible action.
Infinitely repeated games
One-shot games provide an excellent introduction to game theory; however, they can lead to mistaken conclusions when applied too rigidly to the business world. A crucial element missing in one-shot games is time. The business world is characterized by numerous decisions made over an extended period of time. A payoff is associated with each decision, and the players also have memory of past decisions. For all practical purposes, the business time horizon is infinite — the game is never-ending. The result is an infinitely repeated game.
Chapter 12: Game Theory: Fun Only if You Win
In infinitely repeated games, you need to take into account not only how your rival plays this round, but also how this round of the game influences future rounds. Another factor you must consider in infinitely repeated games is the time value of money. A dollar received today is worth more than a dollar received one year from now, because the dollar received today can earn interest. Thus, future payoffs must be adjusted by using the present value calculation I describe in Chapter 1.
The typical strategy used in an infinitely repeated game is the trigger strategy. A trigger strategy is contingent on past play — a player takes the same action until another player takes an action that triggers a change in the first player’s action. An example of a trigger strategy used in games involving a prisoner’s dilemma is tit-for-tat. When you use a tit-for-tat strategy, you start by assuming players cooperate. In any subsequent round, you do whatever your rival did in the previous round. Thus, if your rival cheated on an understanding in the last round, you cheat this round. If your rival cooperated in the last round, you cooperate this round. A tit-for-tat strategy tends to lead to cooperation because it punishes cheaters in the next round. In addition, it forgives cheaters if they subsequently decide to cooperate. One requirement of the tit-for-tat strategy is that the players are stable. The players remember how the game was played in the previous period. New players can upset the necessary balance by not having the required memory of past behavior.
In this example, I talk about two railroads you may recognize from the board game Monopoly — the Pennsylvania Railroad and the B & O Railroad. Here, they’re competing for traffic between the same cities. The railroads can either cheat on one another and charge low prices for freight, or they can cooperate and charge high prices. I show the resulting payoff table of annual profits in Figure 12-5.
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Figure 12-5:
An infinitely repeated game.
In this payoff table, both railroads have a dominant action — to charge a low price. As a result, they both earn $0. This result is yet another prisoner’s dilemma.
If these railroads play the game a long time, they both recognize that cooperating by charging a high price allows both to earn $15 million each and every year. Cheating by charging a low price may lead to a large payoff one year, but the cost associated with having your rival cheat in the next year is very high. Consider what happens in a tit-for-tat strategy:
1. If the B & O Railroad charges a high price, and the Pennsylvania
Railroad cheats and charges a low price, B & O’s losses are $30 million and Pennsylvania’s profit is $35 million. 2. Following a tit-for-tat strategy, the B & O Railroad charges a low price the next year, and the Pennsylvania Railroad continues to charge a low price.
The railroads are locked into a situation of zero profit forever, because the B & O Railroad continues to charge the low price that the
Pennsylvania Railroad charged the previous year. The railroads are in a prisoner’s dilemma.
3. Following a tit-for-tat strategy, the B & O charges a low price the next year and the Pennsylvania Railroad charges a high price.
The B & O Railroad earns $40 million in profit and Pennsylvania loses $25 million. But now the game can return to cooperation. The B & O forgives the Pennsylvania Railroad for cheating in the first round and in future rounds, each railroad earns $15 million.
In some sense, the Pennsylvania Railroad has to accept punishment for cheating in the first place. But accepting that punishment one year leads to a situation where both railroads return to $15 million annual profit. If the railroads continue to cheat by charging a low price, each will recognize that cooperation does not pay, and they will be forever locked into charging a low price and receiving zero profit.
In this infinite game, both railroads make more profit if they cooperate the entire time and never fall into a tit-for-tat strategy.
If you have an infinite time horizon, the present value of a constant stream of future net revenue equals
where π is the net revenue earned each year and i is the interest rate.
