Simultaneous games

Chapter 4: Simultaneous Games
Overview
Interest will not lie.
17th-century proverb
[1]

Softening sales cause both ford and gm to reconsider their pricing.
[2]
If both move at the
same time, then they are playing a simultaneous-move game. Figure 11 presents an
example of a simultaneous-move game. It's important that you understand how to
interpret games like this one, so please read this paragraph very carefully. In this game
Player One chooses A or B, while at the same time Player Two chooses X or Y. Each
player moves without knowing what the other person is going to do. The players'
combined moves determine their payoffs. For example, if Player One chooses A, and
Player Two chooses X, then we are in the top left corner. Player One scores the first
number, 10, as his payoff, and Player Two scores the second number, 5, as her payoff. If
Player One chooses A, and Player Two chooses Y, then we would be in the top right box,
and Player One scores 3 while Player Two scores 0. In all simultaneous-move games
Player One will always be on the left, and Player Two will always be on top. The first
number in the box will usually be Player One's payoff and the second will be Player
Two's payoff. The players always know what score they will receive if they end up in any
given box. The players, therefore, see Figure 11 before they move. Each player knows
everything except what his opponent is going to do.

Figure 11
As with sequential games, in simultaneous games a player's only goal is to maximize his
payoff. The players are not trying to win by getting a higher score than their opponents.
Consequently, Player Two would rather be in the top left box (where Player One gets 10,
and Player Two gets 5) than the bottom right box (where Player One gets 1, and Player
Two gets 4).

What should the players do in a simultaneous game? The best way to solve a
simultaneous-move game is to look for a dominant strategy. A dominant strategy is one
that you should play, regardless of what the other player does. In Figure 11, strategy A is
dominant for Player One. If Player Two chooses X, then Player One gets 10 if he picks A,
and 8 if he picks B. Thus, Player One would be better off playing A if he knows that
Player Two will play X. Also, if Player Two plays Y, then Player One gets 3 if he plays A
and 1 if he plays B. Consequently, Player One is also better off playing A if Player Two
plays Y. Thus, regardless of what Player Two does, Player One gets a higher payoff
playing A than B. Strategy A is therefore a dominant strategy and should be played by
Player One no matter what.
A dominant strategy is a strategy that gives you a higher payoff than all of your other
strategies, regardless of what your opponent does.
Player Two does not have a dominant strategy in this game. If Player Two believes that
Player One will play A, then Player Two should play X. If, for some strange reason,
Player Two believes that her opponent will play B, then she should play Y. Thus, while
Player One should always play A no matter what, Player Two's optimal strategy is
determined by what she thinks Player One will do.
A dominant strategy is a powerful solution concept because you should play it even if
you think your opponent is insane, is trying to help you, or is trying to destroy you.
Playing a dominant strategy, by definition, maximizes your payoff.
To test your understanding of dominant strategies, consider this: Is stopping at a red light
and going on a green light a dominant strategy when driving? Actually, no, it isn't. You
only want to go on green lights and stop on red lights if other drivers do the same. If you
happened to drive through a town where everyone else went on red and stopped on
green, you would be best off following their custom. In contrast, if everyone in this
strange place were intent on electrocuting herself, you would be best served by not
following the crowd. Avoiding electrocution is a dominant strategy; you should do it
regardless of what other people do. In contrast, driving on the right side of the road is not
a dominant strategy; you should do it only if other people also do it.
Let's return to Ford and GM's pricing game. Figures 12 and 13 present possible models

for the auto pricing game. In these games Ford is Player One while GM is Player Two. In
response to weakening sales, both firms can either offer a discount or not offer a
discount. Please look at these two figures and determine how the firms' optimal
strategies differ in these two games.

Figure 12

Figure 13
In Figure 12, offering a discount is a dominant strategy for both firms since offering a
discount always yields a greater profit. Perhaps in this game, consumers will purchase
cars only if given discounts. Figure 13 lacks dominant strategies. If your opponent offers
a discount, you are better off giving one too. If, however, your opponent doesn't lower his
prices, then neither should you. Perhaps in this game consumers are willing to forgo
discounts only as long as no one offers them. Of course, if you can maintain the same
sales, you are always better off not lowering prices. This doesn't mean that neither firm
in Figure 13 should offer a discount. Not offering a discount is not a dominant strategy.
Rather, each firm must try to guess its opponent's strategy before formulating its own
move.
The opposite of a dominant strategy is a strictly stupid strategy.
[3]
A strictly stupid
strategy always gives you a lower payoff than some other strategy, regardless of what
your opponent does. In Figure 12, not offering a discount is a strictly stupid strategy for
both firms, since it always results in their getting zero profits. In a game where you have
only two strategies, if one is dominant, then the other must be strictly stupid.
A strictly stupid strategy is a strategy that gives you a lower payoff than at least one of
your other strategies, regardless of what your opponent does.
Knowing that your opponent will never play a strictly stupid strategy can help you
formulate your optimal move. Consider the game in Figure 14 in which two competitors
each pick what price they should charge. Player Two can choose to charge either a high,

medium, or low price, while for some reason Player One can charge only a high or low
price. As you should be able to see from Figure 14, if Player One knows that Player Two
will choose high or medium prices, than Player One will be better off with high prices. If,
however, Player Two goes with low prices, then Player One would also want low prices.
The following chart shows Player One's optimal move for all three strategies that Player
Two could employ:

Figure 14
Table 1
Player Two's Strategy Player One's Best Strategy*
High High
Medium High
Low Low
*If he knows what Player Two is going to do.
When Player One moves, he doesn't know how Player Two will move. Player One,
however, could try to figure out what Player Two will do. Indeed, to solve most
simultaneous games, a player must make some guess as to what strategies the other
players will employ. In this game, at least, it's easy to figure out what Player Two won't
do because Player Two always gets a payoff of zero if she plays low. (Remember, the
second number in each box is Player Two's payoff.) Playing high or medium always
gives Player Two a positive payoff. Consequently, for Player Two, low is a strictly stupid
strategy and should never be played. Once Player One knows that Player Two will never
play low, Player One should play high. When Player Two realizes that Player One will
play high, she will also play high since Player Two gets a payoff of 7 if both play high and
gets a payoff of only 5 if she plays medium while Player One plays high.
Player Two will play high because Player One also will play high. Player One, however,
only plays high because Player One believes that Player Two will not play low. Player
Two's strategy is thus determined by what she thinks Player One thinks that Player Two
will do. Before you can move in game theory land, you must often predict what other
people guess you will do.

[1]
Browning (1989), 389.
[2]
Wall Street Journal (July 2, 2002).
[3]
Game theorists use the phrase 'dominated strategies,' but this phrase can be confusing
since it looks and sounds like 'dominant strategies.' Consequently I have decided to substitute
the more scientific sounding 'strictly stupid' for the term 'dominated.'
A Billionaire’s Political Strategy
Warren Buffett once proposed using dominant and strictly stupid strategies to get both
the Republicans and Democrats to support campaign finance reform.
[4]
He suggested
that some eccentric billionaire (not himself) propose a campaign finance bill. The
billionaire promises that if the bill doesn’t get enacted into law, then he will give $1 billion
to whichever party did the most to support it. Figure 15 presents this game where each
party can either support or not support the bill. Assume that the bill doesn’t pass unless
both parties support it. The boxes show the outcome rather than each party’s score.

Figure 15
In the game that Buffett proposed, supporting the bill is a dominant strategy, and not
supporting it is strictly stupid. If the other party supports the bill then you have to as well
or else they get $1 billion. Similarly, if they don’t support the reform, you should support
the bill, and then use your billion dollars to crush them in the next election. Buffet’s plan
would likely work, and not even cost the billionaire anything, because both parties would
always play their dominant strategy.
[4]
Campaign for America (September 12, 2000).
More Challenging Simultaneous Games
Games involving dominant or strictly stupid strategies are usually easy to solve, so we

will now consider more challenging games.
Coordination Games
How would you play the game in Figure 16? Obviously you should try to guess your
opponent's move. If you're Player One you want to play A if your opponent plays X and
play B if she plays Y. Fortunately, Player Two would be willing to work with you to
achieve this goal so, for example, if she knows you are going to play A she will play X. In
games like the one in Figure 16 the players benefit from cooperation. It would be silly for
either player to hide her move or lie about what she planned on doing. In these types of
games the players need to coordinate their actions.

Figure 16
Traffic lights are a real-life coordination mechanism. Consider Figure 17. It illustrates a
game that all drivers play. Two drivers approach each other at an intersection. Each
driver can go or stop. While both drivers would prefer to not stop, if they both go they
have a problem.

Figure 17
Coordination games also manifest when you are arranging to meet someone, and you
both obviously want to end up at the same location, or where you're trying to match your
production schedule with a supplier's deliveries. Figure 18 shows a coordination game
that two movie studios play, in which they each plan to release a big budget film over
one of the next three weeks. Each studio would prefer not to release its film when its rival
does. The obvious strategy for the studios to follow is for at least one of them to
announce when its film will be released. The other can choose a different week to
premiere its film, so that both can reap high sales.

Figure 18
Technology companies play coordination games when they try to implement common
standards. For example, several companies are currently attempting to adopt a high-
capacity blue-laser-based replace-ment for DVD players. Consumers are more likely to

buy a DVD replacement if there is one standard that will run most software, rather than if
they must get separate machines for each movie format. Consequently, companies have
incentives to work together to design and market one standard.
In game theory land you do not trust someone because she is honorable or smiles
sweetly when conversing with you. You trust someone only when it serves her interest to
be honest. In traffic games a rational person would rarely try to fake someone out by
pretending to stop at a red light only to quickly speed through the traffic signal. Since
coordination leads to victory (avoiding accidents) in traffic games, you should trust your
fellow drivers. In all coordination games your fellow player wants you to know her move
and her benefits from keeping her promises about what moves she will make. The key to
succeeding in coordination games is to be open, honest, and trusting.
Trust Games
Trust games are like coordination games except that you have a safe course to take if
you're not sure whether your coordination efforts will succeed. Figure 19 illustrates a
trust game. In this game, both players would be willing to play A if each knew that the
other would play A as well. If, however, either player doubts that the other will play A,
then he or she will play B. Playing B is the safe strategy because you get the same
moderate payoff regardless of what your opponent does. Playing A is more risky; if your
opponent also plays A, you do fairly well. Playing A when your opponent plays B,
however, gives you an extremely low payoff.

Figure 19
Figure 20 illustrates a trust game where both you and a coworker demand a raise. In this
game your boss could and would be willing to fire one of you, but couldn't afford to lose
you both. If you jointly demand a raise, you both get it. Alas, if only one tries to get more
money, he gets terminated.