A | B | |

A | 1,1 | 5,0 |

B | 0,5 | 4,4 |

soccer | ballet | |

soccer | 4,2 | 1,0 |

ballet | 0,1 | 2,4 |

Dove | Hawk | |

Dove | 1,1 | 0,2 |

Hawk | 2,0 | -1,-1 |

stay | swerve | |

stay | -50,-50 | 100,0 |

swerve | 0,100 | 50,50 |

A | B | |

A | 100,50 | 0,0 |

B | 0,0 | 50,100 |

Monitor | Not | |

Work | 50,90 | 50,100 |

Shirk | 0,-10 | 100,-100 |

left | right | |

left | -1 | 1 |

right | 1 | -1 |

-4, -4 | 0, -2 | 0, -2 |

-2, 0 | -3, -3 | -2, 0 |

-2, 0 | 0, -2 | -4, -4 |

- the total time somebody worked on household matters (where Beth's time is multiplied by a factor of p), and
- the free time they have between 8:00 and 13:00 .

Rock | Scissors | Paper | |

Rock | 0 | 1 | -1 |

Scissors | -1 | 0 | 1 |

Paper | 1 | -1 | 0 |

Rock | Scissors | Paper | |

Rock | 0 | 2 | -1 |

Scissors | -2 | 0 | 1 |

Paper | 1 | -1 | 0 |

A 3-spinner is a spinner with three equal parts, each with a number on it.
Such a 3-spinner is **balanced** if the three numbers sum up to 0.

The (balanced) 3-spinner game is played by two or more players. Each player brings a (balanced) 3-spinner without knowing what (balanced) 3-spinner the other players have chosen. Then the spinners are revealed and spun. HIER VERSCHIEDENE VARIANTEN---DIFFERENZ, ... The player whose spinner shows the highest number takes $1 from the other players. In case of a tie the win is shared.

**SENATE RACE**(compare [Kockesen])
An incumbent senator decides first whether to run an expensive ad campaign for the next election.
After that, the challenger decides whether to enter the race or not.
The chances for the senator to win are 5/6 with the ad campaign, and 1/2 without.
The value of winning the election is 2, of losing -0.5,
and the cost of the add campaign is 1.
(all values in Million dollars)

**SENATE RACE II**(compare [Kockesen])
An incumbent senator (from a rightist party)
runs agains a challenger (from a leftist party).
They are first choosing a political platform,
leftist or rightist. If both choose the same platform, the incumbent wins,
otherwise the challenger wins. Assume that the value of winning is x
and the value of compromising their political views (by choosing
a platform not consistent with it) is -y. There are different variants,
depending on x and y and also on whether the incumbent has to choose first,
the challenger has to choose first, or whether they both have to choose simultaneously.

**ULTIMATUM GAME(n):**
There is a fixed number n of dollar bills for both players.
Ann makes an offer how to share them, which Beth can either accept or reject.
If she accepts, the bills are divided as agreed upon, if she
rejects nobody gets anything.

**TWO-ROUND BARGAINING(n,k):**
There is a fixed number n of dollar bills for both players.
Ann makes an offer how to share them, which Beth can either accept or reject.
If she accepts, the bills are divided as agreed upon.
If the rejects, n-k dollar bills are taken away and only k remain on the desk.
Then Beth makes an offer how to share these three dollars,
which Ann can accept or reject. If Ann accepts, the bills are divided accordingly,
if she rejects, nobody gets anything.

**MATCHING CHAIRS:**
Ann and Beth can select both 3 chairs from 7 chairs, Two L-chairs
and five O-chairs. One L-chair is worth $300, but if you have a pair
of L-chairs, the pair is worth even $800. O-chairs are less valuable:
One is worth $100, a pair is worth $400, but three of them are worth $900.
Beginning with Ann, Ann and Beth alternate selecting a chair
until each of them has three.

So, on the one hand, the money increases if the players wait, on the other hand the percentage they are allowed to take decreases over time, and moreover the money can decreases when the other player takes some. So the trick is to find the right time to take money.

**NIM(n):**
n stones lie on the desk. Beginning with White, Black and White
alternate to remove either one or two stones from the desk.
Whoever first faces an empty desk when having to move loses.
The winner gets $1, the loser loses $1. What are the best
strategies for the two players?

**MYERSON POKER**
Both Ann and Beth put one dollar in the pot.
Ann gets a card and looks at it privately.
Then Ann either folds, in which case Ann gets the money in the pot if Ann's card is red,
or Beth gets the pot otherwise. Ann can also raise by putting another dollar in the pot.
Now Beth either passes, in which case Ann gets the pot, or Beth meets by
putting one more dollar in the pot. If Beth meets, Ann gets the pot if she has
a red card, otherwise Beth gets the pot.

**LEGISLATORS VOTE:**
Three legislators vote whether they allow themselves a salary rise
of $2000 per year.
Since voters are observing, a legislator would estimate the loss of face by having to vote for a rise
as $1000 per year.

a) What happens if all three vote at the same time?

b) What happens if A has to vote first, then B, then C, and all votes are open.
(This is a variant of a game desribed in [Kaminski].)

How would both players play? [Kaminski]

Governmental Procedure for passing a bill:
The bill first goes to the House (H). If it gets less than 1/2 of the votes, it is rejected.
If it gets at least 1/2 but less than 2/3 it goes to the senate (S).
If it gets at least 2/3, it is reviewed by the president (P).

If the bill is in S, then it must get at least 1/2 of the votes to be reviewed by P,
otherwise it is rejected.

P may veto or accept the bill.

If P accepts, the bill becomes law.

If P vetoes, the veto can be overturned by more than 2/3 both in H and S,
voting sequentially, first H, then S. If the veto is not overturned, the bill
is rejected. If the veto is overturned, the bill becomes law.
[modified from Kaminski]

The next few games are very interesting for experimental games

**Shapely Auction**
Two or more players are bidding for a dollar by increments of 10 Cents. The one with the highest bid has to pay
whatever she was bidding and gets the dollar. The one with the second largest bid gets nothing, but still has to pay
her bid ( a little unfair, but that's the way it is).

**CINQUE-UNO:**
There are two players, and each one gets 5 cards, a "5", a "4", a "3", a "2", a "1".
Then the players reveal one of the cards in their hand simultaneously in rounds.
If both are identical, or if one is "1" and the other "5", both cards
remain on the desk. Otherwise the player who played the higher card takes it back to her hand,
whereas the lower card must remain on the desk. The player having first no card
in her hand after a round (after retaking the higher card) looses.

Poker emerged from the French card game Poque (mispronounced by the Americans) in the beginning of the ninteenth century in New Orleans. The original Poque gave each player three cards out of 32 cards, and used pairs and three of a kind. [McDonald1950]]. It is different to many other games insofar as the betting process takes more space than the process of card handling, actually the betting and bluffing is the essence of the game.

The expectation for players when imitating the bank strategy
is -0.057. The first one to use the recently invented (1944) game theory
on Black Jack was Baldwin in 1956. He showed that expectations of -0.008
could be achieved.
Note that a player using this strategy would still lose
in the long run, but she would lose slower. In 1961, the mathematician Thorpe
apperaed in the Casino scene. He used card counting---realizing what position we are in,
and using different moves for different information sets---and was able to
achieve a positive expectation. He also used small computers, hidden under his clothes,
for the card counting. At that time this was not forbidden yet. Thorpe püublished
a book on his black Jack method, which became rather successful for a mathematics book.
The casinos had to react ........

**LE HER** is played with a 52 cards deck, and the ordering is
A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. Two players, White and Black,
get both a card at which they look without showing it to their opponent.
A third card is put, back up, on the desk. Now White can decide whether she wants
to change cards with Black
(of course without knowing what card Black has).
Only if Black has a King she can refuse the exchange.
Next Black has the opportunity to exchange her card with that lying on the desk.
However, if that card was a King, she has to put it back as well.
Now both players reveal their cards, the one with the higher one wins,
where in case of a tie Black wins.

Von Neumann and Morgenstern discussed in their monograph variants of (Stud) Poker in much detail (pages 186-219). Although these games did not exist, they were not supposed to be toy examples, rather the authors claimed that these versions, though simplified, still would carry many of the characteristics of real Poker. In particular, the results they got shed light on the role of bluffing and why it is necessary.

The authors only concentrated on Poker as a two-players game in their book (although they presented quite a bit of theory of n-person zero-sum games. The reason why they didn't try to extend their analysis to 3 or more players for Poker was maybe that a full solution like in their versions of the 2-player Poker would not have been possible for more players.

Maybe the least severe simplification was their assumption that insted of getting five cards, both players receive only one card ouit of a large deck of cards labeled from 1 to S. Their point of view was that all (2598960) possible hands are linearly ordered from good to bad, all of them are equally likely to get, and for Stud Poker they don't consider the procedure of trying to upgrade your hand. There is a slight flaw in that argument: Seeing your hand, you can exclude some other hands for your opponent, which is not the case if you play it on the cards from 1 to S basis, but the amount of this effect is probably neglectable for large S.

**VNMPOKER(S/m,n,r):**

**VNMPOKER([0,1]/m,n):**

**SIMULTANEOUS VNMPOKER(S/m,n,r)**

The idea is simple for all versions. There are two players, Ann and Beth. Each player randomly gets a card between 1 and S (in the S-version) from a deck of r*S cards, or gets a random number between 0 and 1 (in the [0,1] version). Each player looks at her card but doesn't know the opponent's card. There is a minimum amount of money m the players are betting, but this could be raised to the higher level n. As S and r, the numbers n and m are parameters that could be changed to get different versions of the game, all of them with different strategies. The last difference between the versions is whether the game is simultaneous or not. In the ordinary version Ann starts playing the game by either passing (playing for m) or betting (playing for n).

- If Ann passes, both cards are revealed and the player with the higher card wins the money 2m. In case of a draw---both cards showing the same number---the money is split equally.
- If Ann bets, she increases the money to n. Then Beth has two options,
she can either fold or call.
- If she folds, Ann gets the money m. Beth's card is not revealed in that case.
- If Beth calls, she also increases her EINSATZ to n. Then both cards are revealed again, and the player with the higher card wins the money 2n. Again, in case of a draw the money is split equally.

In the simultaneous version both players decide simultaneously whether they want to bet for m or for n. If one of them bets for n and the other for m, the one daring the higher amount (n) wins m from the other one, regardless what the cards show. If both bet the same amount, the one with the larger card wins n respectively m from the other one, again, no win for draws of identical cards.

Chess is a typical two-player zero-sum (as most )
sequential game
with complete information and no randomness. As such, it has an
optimal strategy that can be found by backwards induction.
However, after 40 moves there are at least 10^{50} postions.
which is a 50-digits number
(see here),
so you can maybe guess how many positions
there are possible in total. With so many positions, the game obviously
resists attacks by even the fastest computers so far for complete analysis.

Go, similiar in many characteristics to chess,
has even 2*10^{170} legal positions
(see here).

Roulette is essentially a one-player game, since the opponent, the bank, doesn's decide at all. Therefore Decision Theory Techniques apply. The game can also be analyzed fairly easily, and it turns out that the expected value (per unit bet) for each move is exactly the same, namely -1/37 for european roulette, and -2/38 for american roulette. It's totally a game of luck, no skill involved.

**CRAZY BETH INVESTMENT (p):**
Ann and Beth invest their $100 either
in bonds, with 6% return, or in in a risky venture.
The venture requires $200 to be a success, then the return
is 10%.
But if the total investment is less than $200, then the
venture is a failure and yields no return. There is no
communication between Ann and Beth.
However, Ann is uncertain about
Beth's preferences. With probability p, Ann believes that
Beth is a little crazy and likes the venture, giving her
satisfaction 120 in each case.
[Kockesen] (compare INVESTMENT above)

**DATING**
(Signalling: Incomplete Information but the possibility to
observe your opponents moves and learn from them)
Adam takes Beth out for a date. She believes the
probabilities of him
being smart or dump are 50% each. Adam wants Beth (2 units
worth)
and tries to look smart.
Being funny "costs" a smart Adam 1 unit, but a dumb Adam 3
units. [Kockesen]

Look at the dating game file, DatingGame.gbt

Example ......

Example ......

Example ......

Example ......

- Levent Kockesen, Extensive Form Games with Perfect Information, http://network.ku.edu.tr/~lkockesen/teaching/uggame/lectnotes/uglect5.pdf
- Branislav Slantchev, Game Theory: Dominance, Nash Equilibrium, Symmetry, pdf script, http://www.polisci.ucsd.edu/~bslantch/courses/gt/04-strategic-form.pdf
- Levent Kockesen, http://network.ku.edu.tr/~lkockesen/teaching/uggame/lectnotes/intro.pdf
- Michael Mesterton-Gibbons, Game Theory, in: Handbook of Discrete and Combinatorial Mathematics (Kenneth H. Rosen ed.). CRC Press, 1999.
- Michael Mesterton-Gibbons, An Introduction to Game-Theoretic Modelling, Addison-Wesley 1992.