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title: Games in Game theory

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# Games in Game theory

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- finite: Game is said to be finite when each player has a finite number of options, the number of players is finite, and the game cannot go on indefinitely.


### Keynes Beauty Contest Game
- Each player names an integer between 1 and 100.
- The player who names the integer closest to two thirds of the average integer wins a prize, the other players get nothing.
- Ties are broken uniformly at random.


### A Prisoner's Dilemma Game

|     |        |        |
| --- | ------ | ------ |
|     | C      | D      |
| C   | -1, -1 | -3, 0  |
| D   | 0, -3  | -2, -2 |

### A Coordination Game

|       |      |       |
| ----- | ---- | ----- |
|       | Left | Right |
| Left  | 1, 1 | 0, 0  |
| Right | 0, 0 | 1, 1  |

### A Matching Pennies Game

|       |       |       |
| ----- | ----- | ----- |
|       | Heads | Tails |
| Heads | 1, -1 | -1, 1 |
| Tails | -1, 1 | 1, -1 |


### A Penalty-Kick Game

|               |       |       |
| ------------- | ----- | ----- |
| Kicker/Goalie | L     | R     |
| L             | -1, 1 | 1, -1 |
| R             | 1, -1 | -1, 1 |


### A Battle of the Sexes Game

|     |      |      |
| --- | ---- | ---- |
|     | B    | F    |
| B   | 2, 1 | 0, 0 |
| F   | 0, 0 | 1, 2 |

### A Hawk-Dove Game

- anti-coordination games

|      |      |      |
| ---- | ---- | ---- |
|      | Hawk | Dove |
| Hawk | 0, 0 | 3, 1 |
| Dove | 1, 3 | 2, 2 |

### A production-choice game

|                 |      |      |
| --------------- | ---- | ---- |
| Firm 1 / Firm 2 | High | Low  |
| High            | 1, 2 | 4, 5 |
| Low             | 2, 7 | 5, 3 |

### An Advertising Game

|                 |        |       |
| --------------- | ------ | ----- |
| Firm 1 / Firm 2 | Not    | Adv   |
| Not             | 16, 12 | 7, 13 |
| Adv             | 13, 7  | 6, 6  |

### A Traffic Game

|               |          |        |
| ------------- | -------- | ------ |
| Car 1 / Car 2 | go       | wait   |
| go            | -10, -10 | 1, 0   |
| wait          | 0, 1     | -1, -1 |


### The Centipede Game (lecture 4-2)
http://gametheory.cs.ubc.ca/centipede?

The centipede game has two players alternate in making decisions. At each turn, a player can choose between going "down" and ending the game or going "across" and continuing it (except at the last node where going “across” also ends the game). The longer the game goes on, the higher the total utility.  However, a player who ends the game early will get a larger share of what utility there is.

![image](https://hackmd.io/_uploads/BJ8ZUzMY6.png)


### The Ultimatum Game (lecture 4-6)
http://gametheory.cs.ubc.ca/ultimatum/basic

- player 1 makes an offer $x \in \{0,1, ...10\}$ to player 2
- player 2 can accept or reject
- 1 gets $(10-x)$ and 2 gets $x$ if accepted
- Both get 0 if rejected

Rejections violate "rationality"? people value equity, or feel emotions


### Five pirates divide loot (Problem Set 4 - 6.)
- Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treacherous and selfish (especially the captain) each wanting to maximize the number of coins that he gets. 
- It is always the captain who proposes a distribution of the loot. All pirates vote on the proposal, and if half the crew or more go "Aye", the loot is divided as proposed. 
- If the captain fails to obtain support of at least half his crew (which includes himself), all pirates turn against him and make him walk the plank. The pirates then start over again with the next most senior pirate as captain (the pirates have a strict order of seniority denoted by A, B, C, D and E).
- Pirates' preferences are ordered in the following way. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins he receives. Finally, each pirate would prefer to throw another overboard in the case of indifference.

What is the maximum number of coins that the original captain gets to keep across all subgame perfect equilibria of this game?
(Hint, work by **backward induction** to reason what the split will be.)