Game Tree and Nash Equilibrium

QUESTION

In 1960 Coca-Cola introduced Sprite, which today is the worldwide leader in the lemon-lime soft drink market and ranks fourth among all soft drinks worldwide.  Prior to 1999, PepsiCo did not have a product that competed directly against Sprite and was trying to decide whether to introduce such a soft drink.  The 1999 annual profits for PepsiCo were $200 million profit and total annual profits at Coca-Cola were $300 million.

If PepsiCo introduced a new lemon-line soft drink, Coca-Cola could purse one of two possible strategies:

(1) PepsiCo could trigger a price war with Coca-Cola in both lemon-lime and cola markets, with resulting annual profits declining to $100 million for each firm.

(2) Coca-Cola could acquiesce.  The net effect of an acquiescence strategy would result in Coca-Cola and PepsiCo earning total profits of $275 million and $227 million, respectively.

Part 1:  Diagram this situation using the extensive or tree-form of a game and identify the Nash Equilibrium

Part 2:  If you were a manager of PepsiCo, would you try to convince your colleagues that introducing the new soft drink is the most profitable strategy? Why or why not?

Part 3:  Let “p” represent the probability that Coke engages in a price war. What is the break-even probability “p” for which Pepsi’s expected profit from introducing Sprite exceeds Pepsi’s profit from not introducing Sprite?

ANSWER

Game Tree and Nash Equilibrium

To analyze this situation using a game tree, we can outline the decision-making process of Coca-Cola and PepsiCo. The nodes in the tree represent decision points, and the branches represent the choices available to each company. Let’s diagram the game tree:

bash
Coca-Cola
/ \
PepsiCo Acquiesce Introduce
/ \ / \
Coca-Cola $275, $227 Price War $100, $100
/ \ / \
$200, $300 $227, $275 $100, $100 $100, $100

In this game tree:

  • The first decision is made by PepsiCo: either to Introduce the new lemon-lime soft drink or not to Introduce.
  • The second decision is made by Coca-Cola: either to Acquiesce or to engage in a Price War.
  • The payoffs for each company are represented as (PepsiCo, Coca-Cola).

The Nash Equilibrium occurs when both companies make their best decisions given the choices of the other. In this case, the Nash Equilibrium is at the point where PepsiCo chooses to Introduce and Coca-Cola chooses to Acquiesce. This is because neither company can improve its payoff by unilaterally changing its decision. Both companies earn profits of $227 million and $275 million, respectively.

Part 2: Managerial Decision for PepsiCo

As a manager of PepsiCo, the decision to introduce the new lemon-lime soft drink hinges on maximizing profits. Looking at the potential outcomes:

  • If PepsiCo introduces the new soft drink and Coca-Cola acquiesces, PepsiCo’s profit is $227 million.
  • If PepsiCo introduces the new soft drink and Coca-Cola engages in a price war, PepsiCo’s profit is $100 million.

Comparing these outcomes to the status quo, where PepsiCo doesn’t introduce the soft drink and earns $200 million, it’s evident that introducing the new soft drink can be beneficial. Even in the worst-case scenario (price war), PepsiCo’s profit is not significantly lower than its current profit.

Moreover, if Coca-Cola acquiesces, PepsiCo’s profit would increase, making the introduction strategy even more appealing. Therefore, I would advocate for introducing the new soft drink. It has the potential to either maintain or improve PepsiCo’s profits, making it a favorable strategic move.

Part 3: Break-Even Probability

To determine the break-even probability “p,” where Pepsi’s expected profit from introducing Sprite exceeds Pepsi’s profit from not introducing Sprite, we need to equate the expected profits from both options:

Expected Profit (Introduce) = Expected Profit (Not Introduce)

Let’s denote the probabilities of Coke engaging in a price war and acquiescence as “p” and “1-p” respectively.

Expected Profit (Introduce) = p * $100 million + (1-p) * $227 million

Expected Profit (Not Introduce) = $200 million

Setting these two equal: p * $100 million + (1-p) * $227 million = $200 million

Solving for “p”: $100 million * p + $227 million – $227 million * p = $200 million $127 million * p = $27 million p = $27 million / $127 million p ≈ 0.2126

Therefore, the break-even probability “p” for which Pepsi’s expected profit from introducing Sprite exceeds the profit from not introducing Sprite is approximately 0.2126, or about 21.26%.

In conclusion, introducing the new lemon-lime soft drink is a strategically sound decision for PepsiCo, as it offers the potential for increased or comparable profits. The break-even probability indicates the threshold at which this decision becomes advantageous. By carefully considering the game theory scenario and the potential outcomes, PepsiCo can make an informed and profitable choice.

 

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