Untied and Air "R" Us are the only two airlines operating...

Untied and Air "R" Us are the only two airlines operating flights between Collegeville and Bigtown. That is, they operate in a duopoly. Each airline can charge either a high price or a low price for a ticket. The accompanying matrix shows their payoffs, in profits per seat (in dollars), for any choice that the two airlines can make. a. Suppose the two airlines play a one-shot game-that is, they interact only once and never again. What will be the Nash (noncooperative) equilibrium in this one-shot game? b. Now suppose the two airlines play this game twice. And suppose each airline can play one of two strategies: it can play either always charge the low price or tit for tat-that is, it starts off charging the high price in the first period, and then in the second period it does whatever the other airline did in the previous period. Write down the payoffs to Untied from the following four possibilities: i. Untied plays always charge the low price when Air "R" Us also plays always charge the low price. ii. Untied plays always charge the low price when Air "R" Us plays tit for tat. iii. Untied plays tit for tat when Air "R" Us plays always charge the low price. iv. Untied plays tit for tat when Air "R" Us also plays tit for tat.

Untied and Air "R" Us are the only two airlines operating flights between Collegeville and Bigtown. That is, they operate in a duopoly. Each airline can charge either a high price or a low price for a ticket. The accompanying matrix shows their payoffs, in profits per seat (in dollars), for any choice that the two airlines can make.
a. Suppose the two airlines play a one-shot game-that is, they interact only once and never again. What will be the Nash (noncooperative) equilibrium in this one-shot game?
b. Now suppose the two airlines play this game twice. And suppose each airline can play one of two strategies: it can play either always charge the low price or tit for tat-that is, it starts off charging the high price in the first period, and then in the second period it does whatever the other airline did in the previous period. Write down the payoffs to Untied from the following four possibilities:
i. Untied plays always charge the low price when Air "R" Us also plays always charge the low price.
ii. Untied plays always charge the low price when Air "R" Us plays tit for tat.
iii. Untied plays tit for tat when Air "R" Us plays always charge the low price.
iv. Untied plays tit for tat when Air "R" Us also plays tit for tat.

Key Concepts

Payoff Matrix

A payoff matrix is a tabular representation of the outcomes for each player based on the combination of strategies chosen. It is a fundamental analytical tool in game theory that allows players to compare potential profits or losses and decide on their optimal strategy.

Tit-for-Tat Strategy

The tit-for-tat strategy is a conditional strategy used in repeated games where a player starts by cooperating (or choosing a high price) and then mimics the opponent's previous move in subsequent rounds. This strategy encourages mutual cooperation by rewarding cooperation and punishing defection, making it a powerful tactic in sustaining cooperative outcomes.

Repeated Game

A repeated game involves playing a stage game (such as a pricing game in a duopoly) multiple times. The repeated structure allows players to develop strategies that can enforce cooperation or punishment, as future interactions can influence current decisions.

Duopoly

A duopoly is a market structure where only two firms operate. In game theory, a duopoly setup simplifies analysis by focusing on the strategic interaction between two players, whose decisions directly affect each other's outcomes. This concept is useful for understanding competitive behavior in limited markets.

Nash Equilibrium

A Nash equilibrium is a solution concept in game theory where no player can increase their payoff by unilaterally changing their strategy, given the strategy of the other player. It reflects a situation of mutual best responses and is a key tool for predicting outcomes in noncooperative games, including one-shot and repeated settings.

One-Shot Game

A one-shot game is an interaction where players make their decisions only once, without the possibility of future retaliation or reward. This setting eliminates the influence of potential future consequences and typically leads to an equilibrium based solely on immediate payoffs.

Transcript

00:05 We have two airlines, air or us and untied, and we've got a payoff matrix for different strategies, low price or high price, and the profit per seat they'll make with the different strategies on a route that they both act as a duopoly own.

00:25 And in part a, we want to know if they only play one time, it's a one -shot game, what would our nash equilibrium be? in that case, each does what's best for it.

00:40 So air or us would say, well, if we charge, if untied charges a low price, then we need to charge a low price too.

00:52 Otherwise, if we charge high, we won't make any profit.

00:56 So that would be their choice.

00:59 If untied charges a high price, then a .r.

01:06 Us would prefer the $50 a seat profit to the 40, and they would also charge a low price in that case, too.

01:16 And then when we look at things from untied's point of view, if a .r.

01:21 Us charges a low price, they could make 20 a c or charging a high price zero.

01:27 So they'll charge the low price.

01:32 And then if air or us charges a high price and they charge a low, they'll make 50.

01:38 If they charge a high, they'll make 40.

01:40 So their dominant strategy is to charge a low price, and that's the same for air or us.

01:47 So our outcome, if it's a one -shot game, they only play once...

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