# Strategy for Information Markets/Background/Nash equilibrium

Game theory is a set of theoretical tools used to predict how people will behave in strategic interactions. The bulk of game theory lies beyond this book, but a grounding in certain aspects of game theory is necessary to appreciate some of the material. In this section, we present some background on games where players move simultaneously. Particularly, we cover public goods which are important for thinking about creation of information goods and coordination games, which are important for studying markets with network externalities.

## Game matrices

Game theory is built around the outcomes and strategies of games. Games include two or more "players" who are affected by the outcome of the game. Each player has a set of possible "moves" that they may take. A game matrix is a table that shows all of the possible moves of each player in a game, and the "payoffs" or rewards that each player will receive from all possible strategies. From a game matrix, one can visualize a given player's strategies, and make predictions of the outcome of a game.

### Example: a public goods game

A public good is one that is non-excludable (no one can prevent others from using it,) and non-rival (there is plenty to go around.) A public goods game is one in which each player can choose how much to invest in a public good when the benefit from that public good goes to all players, regardless of who chose to invest. In the case of information goods, this might be thought of as public broadcasting, where individuals can choose to donate different amounts (or not at all) to public broadcasting, but are able to tune in to the broadcasts in any case.
To make a simple model of it, consider a game in which there are only two players. Each has the option to invest 0 or 4 in a public broadcasting. The quality of broadcasting improves with more investment, with a return of 3/4 of the total investment. So, if $X$ is contributed overall, then each player receives a benefit of ${\frac {3}{4}}X$.
* When only one player invests, that player has a net loss because they paid 4 for the good but only received 3 benefit, while the player who chose not to invest receives a net gain of 3.
* When both players invest, each receives a benefit of 6 making their net gain 2.
* When neither player invests, neither pays or receives any benefit, so their net gain is 0.
We can see these as the payoffs in the game matrix below. Player 1 chooses the strategy that determine which row of the matrix is used, and player 2 chooses the strategy that determines which column of the table is used. The payoffs are listed in order with the row-player's payoff first, followed by the column-player's payoff. For instance, if player 1 chooses to invest 4 and player 2 chooses to invest zero, that will put the players in the lower-left cell of the table which shows the payoffs (-1, 3). That means that if the players choose those strategies, player 1 will get a net benefit of -1 and player 2 will get a net benefit of 3.
| Public Good | Public Good | Player 2 | Player 2 |
| Public Good | Public Good | Invest 0 | Invest 4 |
| Player 1 | Invest 0 | 0, 0 | 3, -1 |
| Player 1 | Invest 4 | -1, 3 | 2, 2 |

## Nash equilibrium

A Nash equilibrium is a situation in which neither player in the game has an incentive to unilaterally change their decision. It is an equilibrium because there is no motivation for a player to change what they are doing.
Notice this does not mean the players are necessarily happy with the outcome. The players might all think that there could be a better outcome if they all changed their strategies, but no single player can improve their own payoff by changing their strategy.
To be a bit more technical, Nash equilibrium is built on the idea of best responses. A player's best response is the best strategy they can choose given the strategy that the other player chose. It is possible (and will be the case in public goods games) that a player's best response is the same no matter what strategy the other players choose. However, it is also possible (and will be the case in coordination games) that a player will have a different best response for each possible strategy the other players might choose.
With the idea of best responses, Nash equilibrium can be restated as:
A Nash equilibrium is a collection of strategies (one for each player) such that each player's strategy is a best response to the other player's strategies.

### Example: a public goods game

In a game matrix like the public-goods game above, the way to find the Nash equilibrium is to ask a series of "what if" questions:
* What if player 2 chose to invest zero? Then player 1's best response would be to invest zero. This can be seen by comparing player 1's payoffs. If player 2 invests zero that means the game will be in the left column for sure. Then player 1 is choosing between a payoff of 0 (if player 1 chooses to invest 0) or a payoff of -1 (if they choose to invest 4).
* What if player 2 chose to invest 4? Then player 1's best response would be to invest zero. Since player 2 is choosing the right column of the table, player 1 is choosing between a payoff of 3 (if they choose to invest 0) or a payoff of 2 (if they choose to invest 4).
* What if player 1 chose to invest zero (or invest 4)? We do the same kind of "what if" reasoning from player 2's point of view. Player 2 thinks about what their best response would be to each possible strategy from player 1. In this game, player 2's best response is always to invest zero.
The Nash equilibrium of this game is therefore the situation where each player invests zero. The Nash equilibrium of this game shows the free-rider problem. The problem isn't that someone gets a benefit without paying the investment. The problem is that, if someone can get the benefit without investing, then it is likely nobody will invest and there will be no benefit for anyone.
This is very relevant for information goods because there is always a concern with information goods that potential creators (writers, inventors, etc.) won't bother to make the investment to create an information good if others can benefit from it without paying.

## Coordination games
### Pure coordination games

In some coordination games, all that matters to the players is that they coordinate. In such games, they aren't concerned about which Nash equilibrium occurs, just that one of them does.
For example, suppose that there is a small island nation which has just put down its first roads. There is no tradition or law yet about which side of the road one is supposed to drive on. If two vehicles are heading toward each other, with one driving on the right (the right-hand side from their perspective) and one driving on the left (from their perspective), they'll have a head-on collision and a very bad day. In the game matrix that appears as payoffs of (-10, -10) for both possibilities of player 1 driving on the right and player 2 driving on the left and vice-versa. If, however, they are both driving on the right-hand side they'll pass each other, and things will go fine. In the game matrix that appears as payoffs of (1, 1) when both players choose the strategy Right. Since what matters is not that they're driving on the right-hand side, but rather that they're driving so as not to crash, the players also receive the (1, 1) payoffs if both drive on the left-hand side.
| Which side to drive on? | Which side to drive on? | Player 2 | Player 2 |
| Which side to drive on? | Which side to drive on? | Left | Right |
| Player 1 | Left | 1, 1 | -10, -10 |
| Player 1 | Right | -10, -10 | 1, 1 |

### Coordination and disagreement

In other coordination games, the players still benefit by coordinating, but they disagree on which equilibrium is preferable. The classic, albeit sexist, example game for this is called the "battle of the sexes". The story is that a man and woman want to have a date, and both will enjoy doing something together more than doing things separately. However, the man would rather the date involve a football game and the woman would prefer the date involves a ballet.
In the game matrix, this is represented as each player choosing either Ballet or Football as a strategy. If they choose different things, they have no date, and they receive payoffs of (0, 0). If they both choose Ballet, they're happier, but particularly the woman is happier, so the man receives a payoff of 1 and the woman receives a payoff of 2. If they both choose Football, they are happy to be together, but the man is happier with a payoff of 2, while the woman receives a payoff of 1.
| Battle of the sexes | Battle of the sexes | Woman | Woman |
| Battle of the sexes | Battle of the sexes | Ballet | Football |
| Man | Ballet | 1, 2 | 0, 0 |
| Man | Football | 0, 0 | 2, 1 |
Let's reframe the same game in terms more appropriate for the subject of information goods. Within a business, it is important that the geeks in engineering can communicate with the beancounters in accounting, but they have different tastes in their mathematical software. The geeks want to use Matlab, while the beancounters want to use Excel. If they each choose their personally-preferred software, they can get some of their private work done, but communication won't happen without a great deal of extra hassle.
In the game matrix, this is shown as the players receiving a payoff of (2, 2) if they choose different math software. If they coordinate on Excel, then the beancounter is very happy with a payoff of 10, while the engineer is pleased that they can get work done, but still less happy and has a payoff of 6. If the coordinate on Matlab, the payoffs are reversed.
| Math software | Math software | Geek | Geek |
| Math software | Math software | Excel | Matlab |
| Beancounter | Excel | 10, 6 | 2, 2 |
| Beancounter | Matlab | 2, 2 | 6, 10 |

### Risky coordination

Sometimes coordinating on a particular equilibrium is obviously better, but there's some risk to it. The classic game for discussing this idea is called the "Stag hunt". The story is that two hunters are going off to find dinner. If they each hunt separately for hares (rabbits), they will each catch one and have dinner. If they hunt together, they can catch a stag (deer) and have a lot of dinners. However, if one goes hunting for a stag alone, that hunter can't catch anything and will go hungry, while the other hunter will still catch the hare.
This is shown in the game matrix as a payoff of (5, 5) if they both choose Stag, and (1, 1) if the both choose Hare. Also, if only one chooses Stag, that one receives a payoff of zero. This game is still a coordination game with two Nash equilibria: (Stag, Stag) and (Hare, Hare).
The special (maybe especially bad) thing about this coordination game is that if one player thinks the other just might fail to choose Stag, the worried player might choose Hare just to be safe. The other player might think the same way, and then they will fall into the less-preferred equilibrium. This is especially true in matrices with high payoffs.
In a study done by the University of Houston it was found that when the payoffs are on a small scale like below players choose the risky option 91.7% of the time. But when they played with larger scale payoffs the risky option was only chosen 69.4% of the time. Patience played a role in the Stag hunt as well. When an iterated test was performed as if the pair went hunting every day risky decisions tended to have better results the further into the experiment they were.
An alternative study was done by George Mason University, which results proved counter-intuitive. Their study showed that risk-aversion, even cognitive ability had a negligible effect on how often participants chose '"Stag,"' and patience had a small to medium effect. It was found in this experiment that the most important factor was that the player was male and agreeable. Males were 14% more likely to result in stag-stag, and agreeable people were 6% more likely.
| Stag hunt | Stag hunt | Player 2 | Player 2 |
| Stag hunt | Stag hunt | Stag | Hare |
| Player 1 | Stag | 5, 5 | 0, 1 |
| Player 1 | Hare | 1, 0 | 1, 1 |
Reframing a similar game in more topic-appropriate terms. Suppose that in 1985 two district offices of a business exchange a lot of mail. They are considering switching to fax machines. If both district offices start using fax machines, then they will save a lot of costs for mailing. If only one office gets a fax machine, they will have gone through the expense and hassle for nothing because they won't be able to exchange faxes with the other.
This is shown in the game matrix as a payoff of (0, 0) if they both choose Mail (since there's no change) and a payoff of (2000, 2000) if they both choose Fax, since they can save a lot by switching to the fax machine. However, if only one switches, they wind up exchanging mail anyway. So they one who stuck with Mail gets a payoff of zero, but the one who gets a fax has a payoff of -300.
| Mail or fax | Mail or fax | Player 2 | Player 2 |
| Mail or fax | Mail or fax | Fax | Mail |
| Player 1 | Fax | 2000, 2000 | -300, 0 |
| Player 1 | Mail | 0, -300 | 0, 0 |

## References

* ↑ Game theory is heavily used by economists, but is also used by people studying military strategy, politics, family behavior, and many other fields of human interaction. It is also sometimes used to study other areas, including evolutionary biology and computer engineering.
* ↑ (http://www.uh.edu/econpapers/RePEc/hou/wpaper/2006-01.pdf)
* ↑ http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1764272