Lecture Notes 10: Examples and Discussion of Sequential Games
Order of Moves
An interesting question in sequential games concerns the order of moves. Is it better to move first or is it better to let the other player move first? The answer is basically that there’s no general answer. It depends on the structure of the game. Sometimes it’s better to be the first-mover. Sometimes it’s better to let the other guy move first.
First-Mover Advantage
Two restaurants with similar food are considering whether to set up shop in a mall. Profit is zero for a restaurant that doesn’t enter the market. If only one restaurant enters, then the payoff for that restaurant is 10. But if both restaurants enter, then the market is oversaturated and both earn a profit of −5. The game is played sequentially, with firm 1 first deciding whether to enter. Firm 2 observes this decision and then decides whether to enter.
The game tree for this interaction is shown below.
This game has an obvious first-mover advantage. In the SPE, firm 1 enters the market, knowing that firm 2’s rational response is to stay out.
Second-Mover Advantage
Imagine matching pennies played in sequence. Two players each choose 𝐴𝐴 or 𝐵𝐵. If the strategies match, player 1 gains $1 and player 2 loses $1. If the strategies do not match, player 1 loses $1 and player 1 gains $1. The game is played sequentially, with player 1 choosing his strategy first. Player 2 observes this choice before deciding on her strategy.
This game has an obvious second-mover advantage. It doesn’t matter what player 1 chooses, because player 2 can always follow it up by making sure that the strategies don’t match, guaranteeing a win for player 2.
Here are some real-world scenarios that create a second-mover advantage.
• You are on a committee where you and a coworker are each arguing for a preferred proposal. Most people prefer to speak second so that they can tailor their argument (strategy) in response to what the first person said.
• In political campaigns, it’s usually better for a strong candidate to match a clean campaign with a clean campaign or to launch a dirty campaign if his opponent launches a dirty campaign. Thus, it’s better for him to wait and see what his opponent is going to do.
• In dating, many people prefer to wait and see if the other person is interested in “going further” before making a proposition themselves. Not matching can be embarrassing and costly.
Converting Simultaneous to Sequential Moves
What’s the impact on the players if we convert a simultaneous game to a sequential game with the same payoffs? Again, there is no single answer. It all depends on the structure of the game.
• The games could have the same outcome. In a prisoner’s dilemma, it doesn’t matter if the game is played sequentially or who moves first. Both players have a strictly dominant strategy to confess regardless of the other person’s action.
• It could create a first-mover advantage. The game of chicken has multiple equilibria. But, if it is played sequentially, the first-mover could commit to going straight, which forces the second mover to adapt.
• It could create a second-mover advantage. In matching pennies, the Nash Equilibrium in the simultaneous game is a mixed equilibrium where both players get an average payoff of zero. But if matching pennies were played sequentially, the second player could just adapt to whatever the first player did to guarantee a win.
• It could be better for both players. Consider the examples below. When the game is played simultaneously, the Nash Equilibrium is (𝐵𝐵, 𝐷𝐷) and the equilibrium payoffs are (2,2). When the game is played sequentially, the SPE is (𝐴𝐴, 𝐶𝐶𝐷𝐷), which leads to a payoff vector of (3,4). Interestingly, notice that player 1 chooses 𝐴𝐴 in the sequential game, which is a strictly dominated strategy in the simultaneous game.
C D
A 3,4 1,3
Some Examples
Here are a few interesting examples of applying backwards induction to situations involving strategy and sequential moves.
Foreign Relations
Suppose that Chad (relatively weak) is in danger of being attacked by Libya. But if Libya commits troops to a war with Chad, then it will be in a weakened position against Egypt, who could attack Libya. The presence of Egypt indirectly helps Chad. This is just an example of the old adage that “the enemy of my enemy is my friend”.
But what if we add in Egypt’s rivalry with Israel? Now, if Libya attacks Chad, then Egypt cannot attack Libya because Egypt itself would be in a weakened position against Israel. Thus, the presence of Israel indirectly hurts Chad. How would we update the adage? I guess “the enemy of my enemy’s enemy is my enemy”. You can see that the deciding factor here is whether the number in the circuit is odd or even.
Evading the Draft
The government wants to force citizens to sign up for the military draft, but it could not possibly punish everyone if nobody obeys the order. Instead, the government announces it will punish offenders in alphabetical order. All the people with last name “Aaron” know that they’d better sign up. Knowing that the Aarons sign up, everyone in the “Abington” family will sign up, so then everyone in the “Ackerman” family needs to sign up, etc…
This is similar to shooting the first people in line at a protest. It makes people disperse pretty quickly.
The Three-Way Duel
Albert, Barbara and Chester are engaged in a three-way duel. Albert only hits his target 30% of the time. Barbara hits her target 70% of the time. Chester is a perfect shot. Each is given one bullet, and they shoot in order – Albert, then Barbara and then Chester. If anyone is still alive, then there is a second round. Whom should Albert aim at?
Believe it or not, Albert is better firing up in the air and not aiming at anyone. Killing Barbara is a bad idea because then Chester is going to kill him. But killing Chester isn’t a great idea either. If Albert manages to kill Chester, then Barbara is going to shoot at him instead of shooting at Chester! Better to leave Chester alive and let Barbara use her bullet to shoot at Chester.
up. Strong candidates are going to go after other strong candidates. In this sense, weak people can have a kind of advantage because others don’t behave as aggressively towards them.
Path Consistency in Decision-Making: Playing against Yourself?
Why do people do things like overeat, start smoking and spend all their money when they know it’s going to leave them worse off in the future? Some economists have interpreted this as playing a game against your future self. Things like smoking or overeating create a flow of positive utility in the present, but they create a stock of negative utility the more they accumulate. Our “present self” gets utility from them but then we promise that our “future selves” are going to behave differently. But if you can’t overcome the temptation to overeat today, why should you assume that you’re going to do so in the future?
Declining Industries
An interesting observation is that the largest firm is usually the first to exit a declining industry. Why is this the case? Here is a simple example of a declining industry with two firms that could keep operating for 15 more periods. The table below shows monopoly profits if the small firm operates alone, monopoly profits if the large firm operates alone and the duopoly profit earned by each firm if both are in the market. Each period, the firm chooses whether to exit. Once it exits, it’s gone for good and can’t reenter.
Period Small Firm Alone
Large Firm
Alone Duopoly
1 300 200 50
2 275 175 25
3 250 150 0
4 225 125 −25
5 200 100 −50
6 175 75 −75
7 150 50 −100
8 125 25 −125
9 100 0 −150
10 75 −25 −175
11 50 −50 −200
12 25 −75 −225
13 0 −100 −250
14 −25 −125 −275
Note that the large firm is less profitable operating alone. The idea is that the large firm has high fixed costs associated with being in the market each period. To see what happens, let’s solve this game backwards.
• The small firm, even if alone, will exit in period 13.
• The worst case scenario is for the large firm to operate until period 9; it would never operate beyond this point. Thus, the small firm is guaranteed $250 of monopoly profits from period 9 onward if it can hang on until period 9 (100 + 75 + 50 + 25).
• If the small firm is still in the market in period 7, it will hang on. Even if the large firm is still in the market, the $225 duopoly losses for the small firm earned in periods 7 and 8 (−100 + −125) are more than offset by the $250 profits from period 9 onward.
• Knowing the above, the large firm realizes that – if both are still in the market – the small firm has an incentive to hang on through periods 7 and 8. This guarantees duopoly losses for the large firm. Thus, if it were in a duopoly, the large firm would exit in period 7.
• The small firm figures out that – if the market is still a duopoly by that point – the large firm will exit in period 7. This effectively means that the small firm has a guaranteed monopoly from period 7 on, bringing in total profits of $525. This is more than enough to offset any duopoly losses that it might incur before period 7. Thus, the small firm will definitely hang on.
• The large firm realizes that the small firm is going to hang on. Thus, it might as well exit in period 3 as soon as the duopoly profits turn negative. It knows that the small firm is going to hang on, so there is no point in incurring losses.
Thus, the outcome in this game is that the large firm leaves in period 3 and leaves the small firm with monopoly power from then on. Basically, the small firm can use the fact that the large firm has big fixed costs and incurs bigger losses to its advantage. It can bully the large firm out early because it can afford to hang on for longer. The large firm realizes this and so it’s futile for them to stay in the market after duopoly profits turn negative.
General Results on Subgame Perfect Equilibrium
Here are two important results that hold generally for sequential games. This pair of results is collectively known as Zermelo’s Theorem.
• Every finite game of perfect information has a SPE that can be found using backwards
induction.
• If every payoff for every player is different, then the SPE is unique.
Parlor Games
Here are a few well-known games that fall into the category of finite, sequential games of perfect information. Zermelo’s Theorem applies to these games, and so all have a subgame perfect equilibrium that we should be able to find if we work hard enough.
Tic-Tac-Toe
At his first move, player 1 has 9 choices. After this, player 2 has 8 choices, leading to 72 nodes in the second round of play. After this, player 1 has 7 choices, implying that there are 504 nodes in the third round of play, etc…
Solving by hand probably isn’t possible, but you can program a computer to do it. It has been proven that properly-played tic-tac-toe (i.e. where players choose their SPE strategies) always ends in a tie. Even smart kids can figure this out so that their games will always end in a tie. How? Well, the size of the strategy set is very large but the game is simple and many of these nodes are basically identical.
Chess
Chess is a finite game of perfect information. The rules are fixed; nothing is left to chance; all information is known. Thus, Zermelo’s Theorem proves that it has a SPE which can be found using backwards induction. Can we find it?
Well, even though we can prove that the SPE exists, we can’t actually find it with our current technology. Chess is very complicated and the total number of possible moves is about 10120. Even with our fastest supercomputer, this would take about 10100 years to solve. Unfortunately, the earth is finished in fewer than 1010 years.
Checkers
Checkers is considerably simpler than chess. Cho and Schaeffer went through the backwards induction and proved in 2007 that perfectly played checkers always ends in a tie. Lest you think this is a simple task, their computer program had been in development since 1989!
So Cho and Schaeffer actually computed, explicitly, the optimal strategies for checker. Will this make checkers uninteresting? Probably not. The strategy is too complicated for any human to memorize and use in a game. Again, just because Zermelo’s Theorem tells us that an SPE exists, this does not mean that we can actually find it (chess) or that it’s practical to use it (checkers).
Some Experimental Evidence
Two of the games we looked at in the previous section have been extensively studied in laboratory settings.
Ultimatum Game
Suppose that player 1 is given $100 and told to propose a split with player 2. If player 2 accepts the split, the money is split as specified. If player 2 rejects the split, both get nothing. Game theory gives a very clear prediction of play. Player 2 should accept any offer that beats 0, so player 1 should offer $1 to player 2 and keep the other $99 for himself.
Laboratory play consistently contradicts the SPE prediction for ultimatum games. Splitting the money 50/50 is the most common proposal and player 2 almost always rejects offers worse than 75/25, and sometimes rejects offers as favorable as 60/40.
How to interpret these results is a topic of contention. One explanation is that people place a psychological value on fairness, so the monetary payoff is not the true payoff for the game. There are mixed results on whether larger amounts of money impact the result.
Centipede Game
In the centipede game, the unequivocal SPE is for player 1 to stop immediately. But both players can end up with more money if they behave in a way that is contradictory to the SPE strategies and they keep going for awhile. People often do, and it’s not even unusual for the last player to keep going at the very last stage of the game.
Problems
1. The trust game: Player 1 is given $10 and player 2 is given nothing. Player 1 can “give back” $𝑥𝑥 of his $10. If he does so, player 1 loses 𝑥𝑥 but player 2 gains 3𝑥𝑥. Player 2 then has an opportunity to give a gift of $𝑦𝑦 to player 1, after which the game is over.
a. What values of 𝑥𝑥 and 𝑦𝑦 are chosen in the subgame perfect equilibrium?
b. Now suppose that, after the game is over, player 1 has an opportunity to punch player 2 if he is unhappy with the gift that player 2 gave him. Punching lowers player 1’s payoff by 1 and lowers player 2’s payoff by 5. What values of 𝑥𝑥 and 𝑦𝑦 are chosen in the SPE of this new game?
c. Return again to the scenario in (b) but suppose that player 1’s payoff actually rises by 1 when he punches player 2. Player 2 still loses 5 by being punched. What values of 𝑥𝑥 and 𝑦𝑦 are chosen in the SPE of this game?
2. Consider a two-player game where player 1 chooses how to split a total sum of 100. Let 𝑥𝑥 be the amount going to player 1 and let 𝑦𝑦 be the amount going to player 2. Player 1 proposes the division (𝑥𝑥, 𝑦𝑦). Player 2 either accepts this division or rejects it, in which case the payoff for both players is equal to 0.
a. Suppose the payoff functions are just Π1 = 𝑥𝑥 and Π2 = 𝑦𝑦. What offer (𝑥𝑥, 𝑦𝑦) is made by player 1 in the SPE of this game?
