Ultimatum Game
The ultimatum game is an interesting game that has been widely studied
by researchers in different fields, such as economics and sociology. This is
a non-zero sum, two player, extensive form game with complete information. The
rules of the game are as follows:
- Player 1 is promised 10 dollars from the sponsor of the game.
However, Player 1 must split this money with Player 2, who also knows the
value of the endowment. The amount Player 1 receives depends on the action of Player 2.
- Player 1 decides on the value of an offer to make to Player 2.
- Player 2 can accept or reject the offer.
- If Player 2 accepts the offer, Player 2 receives the offered amount and
Player 1 keeps the remainder.
- If Player 2 rejects the offer, none of the players receive anything.
Below is the illustration of the game tree.
How would you play this game with a stranger?
- Suppose you are player 1. What offer would you make to Player 2? Write this down on a piece of
paper.
- Now suppose you are player 2. What offers would you accept from Player 1? Write this down on a
piece of paper.
- If you were playing against yourself, what would be the payoffs for Player 1 and Player 2?
Now let's do a more game-theoretic analysis of this game. As an exercise, answer the following questions:
- What is the strategy set for Player 1 and for Player 2. How many strategies does each player have?
- What is the set of Nash equilibria of this game. How many equilibria are there?
- What is the set of Subgame Perfect Nash equilibria. How many equilibria are there?
- Give a few Nash equilibria of the game that correspond to "incredible threats".
You now know what game theory prescribes to the players of this game. With this in mind, answer the
following questions:
- How does the game-theoretic solution compares with your initial intuition of the game?
- If the solutions differ, why do you think this is the case?
- Would you play the game-theoretic solution with a stranger?
So here is your chance to play this game with a stranger! Here is how this will work:
- Form a "team" (which will correspond to a player in the game) of small group of students
attending the seminar.
- Send in your team name and the list of team members (name and email address) to daniel.figueiredo@epfl.ch. Also, please
have all team members subscribe to the class mailing list (details in the course webpage). Please register your team by Tuesday, March 28.
- Your team will be randomly selected to play either as Player 1 or as Player 2. Your team will be informed of this decision within one day of registration.
- Your team will be randomly paired with another team, which will be your opponent in the game. At this
point, no teams have knowledge of the identity of their opponents.
- Send in your strategy (just one, either for Player 1 or for Player 2, according to which player
your team is playing) to daniel.figueiredo@epfl.ch. Make sure this strategy is discussed is agreed upon your team members. Of course, your team can play any strategy it wants. There is no right or wrong strategy!
- Please submit your strategy by Wednesday, March 29.
We will discuss the outcome of this experiment in the next lecture. In particular, we will reveal the team
pairings and the payoff to the respective teams. Be prepared to support your point of view and the strategy
taken by your team (whatever that might be).
Good luck!
Teams, Players and Strategies
| Team Name | Team Members | Player
number | Strategy |
| EPFL-1 | Daniel Figueiredo (daniel.figueiredo(at)epfl.ch)
Gianluca Rizzo (gianluca.rizzo(at)epfl.ch) | 1 | Offer 3 |
| The Untouchables | Carlo Caruso (car_car(at)inwind.it)
Matteo Lauriano (matteo.lauriano(at)libero.it) | 1 | Offer 3 |
| FIREWALL | Floriana Vaglica (floriana_vaglica(at)yahoo.it)
Giuseppe Cannella (cannella.giuseppe(at)libero.it) | 2 | Accept 1 or more |
| UFRJ-LAND-1 | Edmundo de Souza e Silva (edmundo(at)land.ufrj.br)
Antonio Augusto de Aragao Rocha (arocha(at)land.ufrj.br) | 2 | Accept 4 or more |
| UFRJ-LAND-2 | Guilherme Dutra Gonzaga Jaime (gdjaime(at)land.ufrj.br)
Hugo Sato (hugosato(at)land.ufrj.br)
David Boechat (davidboechat(at)ufrj.br) | 1 | Offer 5 |
| ZUNZARI | Marchese Vincenzo (zurrro(at)yahoo.it)
Buttitta Stefano (stefano_buttitta(at)virgilio.it)
Tritico Leonardo (triticol(at)caltanet.it) | 2 | Accept 4 or more |
| UTOV | Francesca Lo Piccolo (francesca.lopiccolo(at)tti.unipa.it)
Francesca Martire (francesca.martire(at)uniroma2.it)
Simone Teofili (simone.teofili(at)uniroma2.it) | 1 | Offer 4 |
| Pippo | Giovanni Neglia (giovanni.neglia(at)tti.unipa.it)
Pietro Cassarà (pietro.cassara(at)tti.unipa.it) | 2 | Accept 1 or more |
| BH | Boulat A. Bash (boulat(at)cs.umass.edu)
Honggang Zhang (honggang(at)newworld.cs.umass.edu) | 2 | Accept 3 or more |
| XYZZY | Boris Margolin (margolin(at)cs.umass.edu)
M.S. Raunak (raunak(at)cs.umass.edu) | 1 | Offer 1 |
| Angel | Gagliano Margherita (marge_g(at)libero.it)
Maria Barbara (trinity_82(at)virgilio.it)
Occorso Marinella (marina.occ(at)tiscali.it) | 1 | - |
| Parsifal | Marco Pavone (pavone(at)unipa.it) | 2 | - |
Average offer: 16/5 = 3.2
Average acceptance: 13/5 = 2.6
Results of Single Random Pairing
Player 1
Team name | Player 2
Team name | Payoff |
EPFL-1
Offer 3 | ZUNZARI
Accept 4 or more | 0, 0 |
The Untouchables
Offer 3 | UFRJ-LAND-1
Accept 4 or more | 0, 0 |
UFRJ-LAND-2
Offer 5 | FIREWALL
Accept 1 or more | 5, 5 |
UTOV
Offer 4 | BH
Accept 3 or more | 6, 4 |
XYZZY
Offer 1 | Pippo
Accept 1 or more | 9, 1 |
Winning Team playing as Player 1: XYZZY (Offer 1)
Winning Team playing as Player 2: FIREWALL (Accept 1 or more)
Results of All Possible Pairings
| FIREWALL
Accept 1 or more | UFRJ-LAND-1
Accept 4 or more | ZUNZARI
Accept 4 or more | Pippo
Accept 1 or more | BH
Accept 3 or more | P1 Average |
EPFL-1
Offer 3 | 7, 3 | 0, 0 | 0, 0 | 7, 3 | 7, 3 | 4.20 |
The Untouchables
Offer 3 | 7, 3 | 0, 0 | 0, 0 | 7, 3 | 7, 3 | 4.20 |
UFRJ-LAND-2
Offer 5 | 5, 5 | 5, 5 | 5, 5 | 5, 5 | 5, 5 | 5.0 |
UTOV
Offer 4 | 6, 4 | 6, 4 | 6, 4 | 6, 4 | 6, 4 | 6.0 |
XYZZY
Offer 1 | 9, 1 | 0, 0 | 0, 0 | 9, 1 | 0, 0 | 3.60 |
| P2 Average | 3.20 | 1.80 | 1.80 | 3.20 | 3.0 | 4.60, 2.60 |
Winning Team playing as Player 1: UTOV (Offer 4)
Winning Team playing as Player 2: FIREWALL and Pippo (Accept 1 or more)
Repeated Game Play
We will now play the ultimatum game as a repeated game! The game will proceed as follows. Player 1 will make an
offer to Player 2. This offer will be done via email: after having decided their offer, one member of Team 1 will
send an email to all members of Team 2 (and cc me, Daniel) with the offer. Team 2 analyzes the offer and
then decides either to accept or reject it. This is also done via email: one member of Team 2 will send an email
to all members of Team 1 (and cc me, Daniel) with the accept/reject notice. This exchange concludes one round of
the repeated game. The next round starts by repeating the process above, with Player 1 making an offer, and so on.
We will play 10 rounds!
Please use the text "[ultimatum game]" in the subject of all your emails. This will facilitate identifying game
related emails. Also, please note that no other communication is allowed. The message from Player 1 should contain
just a number (which is the offer being made), while the message from Player 2 should contain just accept/reject
(either accepting or rejecting the offer). Please keep track of your game by using the table below:
| Round |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Team 1
(offer): | | | | | | | | | | |
Team 2
(A/R): | | | | | | | | | | |
Payoff
(this round): | | | | | | | | | | |
Cumulative
Payoff: | | | | | | | | | | |
Finally, the team pairings are the same as before. For clarity, below is the table with the role of your team (either
Player 1 or Player 2) and your opponent. The team members as well as their emails are found in a table above.
Player 1
Team name | Player 2
Team name |
| EPFL-1 | ZUNZARI |
| The Untouchables | UFRJ-LAND-1 |
| UFRJ-LAND-2 | FIREWALL |
| UTOV | BH |
| XYZZY | Pippo |
| Angel | Parsifal |
You can start to play the game. Player 1, send your offer to Player 2!
Good luck!
Results of Repeated Game
|
Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| EPFL-1
(offer): | 3 | 3 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 3 |
| ZUNZARI
(A/R): | A | R | R | R | A | A | A | A | A | A |
| Payoff
(this round):
| 7,3 | 0,0 | 0,0 | 0,0 | 6,4 | 6,4 | 6,4 | 6,4 | 6,4 | 7,3 |
| Cumulative
Payoff:
| 7,3 | 7,3 | 7,3 | 7,3 | 13,7 | 19,11 | 25,15 | 31,19 | 37,23 | 44,26 |
|
Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| The Untouchables
(offer): | 2 | 4 | 1 | 3 | 2 | 2 | 3 | 5 | 3 | 4 |
| UFRJ-LAND-1
(A/R): | R | A | R | R | R | R | R | A | R | A |
| Payoff
(this round):
| 0,0 | 6,4 | 0,0 | 0,0 | 0,0 | 0,0 | 0,0 | 5,5 | 0,0 | 6,4 |
| Cumulative
Payoff:
| 0,0 | 6,4 | 6,4 | 6,4 | 6,4 | 6,4 | 6,4 | 11,9 | 11,9 | 17,13 |
|
Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| UFRJ-LAND-2
(offer): | 2 | 3 | 4 | 4 | 4 | 3 | 3 | 3 | 4 | 4 |
| FIREWALL
(A/R): | R | R | R | A | A | R | R | R | A | A |
| Payoff
(this round):
| 0,0 | 0,0 | 0,0 | 6,4 | 6,4 | 0,0 | 0,0 | 0,0 | 6,4 | 6,4 |
| Cumulative
Payoff:
| 0,0 | 0,0 | 0,0 | 6,4 | 12,8 | 12,8 | 12,8 | 12,8 | 18,12 | 24,16 |
|
Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| UTOV
(offer): | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 3 | 3 |
| BH
(A/R): | R | R | A | A | A | A | A | R | R | R |
| Payoff
(this round):
| 0,0 | 0,0 | 5,5 | 5,5 | 5,5 | 5,5 | 5,5 | 0,0 | 0,0 | 0,0 |
| Cumulative
Payoff:
| 0,0 | 0,0 | 5,5 | 10,10 | 15,15 | 20,20 | 25,25 | 25,25 | 25,25 | 25,25 |
|
Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| XYZZY
(offer): | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 1 |
| Pippo
(A/R): | R | R | R | R | A | A | A | A | A | R |
| Payoff
(this round):
| 0,0 | 0,0 | 0,0 | 0,0 | 5,5 | 5,5 | 5,5 | 5,5 | 5,5 | 0,0 |
| Cumulative
Payoff: | 0,0 | 0,0 | 0,0 | 0,0 | 5,5 | 10,10 | 15,15 | 20,20 | 25,25 | 25,25 |
|
Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Angel
(offer): | 5 | 4 | 4 | 3 | 5 | | | | | |
| Parsifal
(A/R): | A | R | R | R | A | | | | | |
| Payoff
(this round):
| 5,5 | 0,0 | 0,0 | 0,0 | 5,5 | | | | | |
| Cumulative
Payoff:
| 5,5 | 5,5 | 5,5 | 5,5 | 10,10 | | | | | |
Winning Team playing as Player 1: EPFL-1 (Payoff 44)
Winning Team playing as Player 2: ZUNZARI (Payoff 26)
Comments
- All teams lost some rounds in order to reach a "fair" split of the endowment, with Player 2 rejecting several offers
in the first rounds of the game. There was no game with an agreement in a "fair" split from round 1.
- One pair of teams never reached a "fair" split, with Player 1 changing its offer and Player 2 changing its strategy
(The Untouchables x UFRJ-LAND-1). This lead to the worst payoff for both players among all games!
- Some teams playing as Player 1 changed strategy after an implicit "fair" split had been reached. This seems to be the
case for UFRJ-LAND-2 x FIREWALL and UTOV x BH.
- Winning team (for Player 1) reached a "fair" split earlier in the game (in round 5) and kept agreement until the last round (EPFL)
- Winning team (for Player 2) accepted an unequal split as being a fair split early in the game (in round 5) (ZUNZARI)
- In general, high offers were rejected. Out of 14 offers of 3, 12 were rejected (the only two accepted were by the winning
team as Player 2, ZUNZARI). Out of 19 offers of 4, 9 were rejected (with teams BH, Pippo and Parsifal consistently rejecting offers
of 4). Even one offer of 5 was rejected (by team BH)! This may reflect the fact the payoffs had no real value. Therefore, Player 2
was deciding to accept/reject offers based mainly on the grounds of fairness. However, if the payoffs were thousands of dollars
(say offering 4 is offering 4 thousand dollars), Player 2 would probably not have rejected high offers of 3 and 4.
- All teams playing as Player 1 thought that the game gave them an advantage over Player 2, which was reflected by trying to
make offers below 5.
- Two teams rejected the final offer (BH and Pippo). Knowing that the game would end in that round, they still
refused the offer. Moreover, team BH rejected an offer of 3 in the last round.
Last Updated: May 24, 2006 by Daniel Figueiredo.
Please send your comments to daniel.figueiredo@epfl.ch.