ASSIGNMENT: Introduction to game theory and negotiation

Date:

Name of the group:

Names of the members:

University:

Part a

A game in extensive form

Consider a game in normal form:

		Player 1
		L	R
Player 2	T	2,1	-1,-1
	B	-1,-1	1,2

Present the game in extensive form and point out the Nash equilibria of the game.

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The problem of the commons

Consider the problem of the commons and assume that the payoff functions for the players are
u₁(g₁,g₂)=(1-g₁-g₂)g₁
u₂(g₁,g₂)=(1-g₁-g₂)g₂,
where player 1 chooses g₁ and player 2 chooses g₂.

Solve the Nash equilibrium for the game.

Solve a Pareto optimal solution for the game by the method of improving directions. Use the Nash equilibrium as the initial tentative agreement. How many iterations do you need to take to find the Pareto optimal solution?

· The steps in the method:

1. Evaluate the unit gradients of the payoff functions

2. Solve the bisecting direction; that is the sum of the unit gradients

3. Find the most preferred alternatives for the players in the direction

4. Choose a new tentative agreement and check if it is Pareto optimal; if not go to step 1

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Part b

In part b you negotiate interactively by the Joint Gains software and solve the problem of buyer and seller. Follow the instructions below; there are video clips presenting the use of Joint Gains at steps 4, 5 and 6. The questions to be answered are presented in steps 7 and 8. To work with Joint Gains you need a Java enabled web browser.

Note:

q You should be familiar with the case of buyer and seller .

q If you do not have a pair, you may work alone by playing the roles of both parties.

1. Starting Joint Gains

· Start the Joint Gains

o Click “start Joint Gains” link in the Joint Gains front page

2. Creating the case

· The group is the facilitator who creates the negotiation case in the Joint Gains software

· The members of the group are the participants (parties) in the negotiation

· First the facilitator clicks “new” in the “case” menu and enters

o case name

o case facilitator name

o password for the facilitator

· The facilitator adds the participants by clicking “add” and enters

o their public names, e.g., buyer and seller. They are visible for the other participant

o user ids that are used when the participants open the case at step 5

o passwords for the participants

· The facilitator adds the decision variables (issues), delivery time and price, by selecting the “decision variables” tab and clicking “add”

o Use the following values:

Decision variable	Minimum value	Maximum value
Time	0 days	12 days
Price	0 €	400 €

· The facilitator can publish the case by clicking “publish” in the “case” menu

· See also the video clip, instruction 4

3. Creating the sessions

· In this assignment you negotiate starting from two different initial tentative agreements

o two sessions are needed

· Joint Gains adds one session by default so the facilitator needs to add the other one by clicking “add session” in the “case” menu

· Now there are two session windows

· The facilitator starts the sessions separately by clicking “start” in the “session” menu and enters

o the initial tentative agreement

o the resolution parameters r_time and r_price that define the size of the circle (ellipse) on which the participants search for their most preferred alternative

· Use the following default values:

	initial value for time	initial value for price	r_time	r_price
Session 1	2 days	200 €	0.5 d	5 €
Session 2	10 days	200 €	0.5 d	5 €

· After the sessions are started the facilitator can close the case and session creation by clicking “close” in the “case” menu

4. Video clip on the case and session creation

· There is a video clip illustrating how to create the case, add a session and start it in the Joint Gains

· Video clip: Setting up the negotiation case in Joint Gains, 3:45 mins

o with sound (avi 3067kb)

o no sound (avi 600 kb)

o animation (gif 362 kb)

5. Negotiation steps

· Each participant opens the case by selecting “open” in the “case” menu and entering

o user id

o password

o case name

· Note that you can work on separate workstations simultaneously

· It is recommended that the participants negotiate the session 1 first and after finishing it complete the session 2

· The negotiation consists of three tasks:

1. Choose the best alternative on the ellipse

2. Choose the most preferred alternative in the compromising direction

3. Accept or do not accept the proposal

· If both participants accept, the proposal is the new tentative agreement; go to task 1

· If either of them does not accept, the negotiation ends

· To negotiate click “continue” in the “session” window and answer the questions until you are indifferent; then click “indifferent”.

o Wait until the other participant has completed the task and click “continue” button to go to the next task

· Video clip: Choosing the best alternative on the ellipse by Joint Gains, 1:31 mins

o with sound (avi 1323 kb)

o no sound (avi 318 kb)

o animation (gif 212 kb)

6. Viewing the results

· You can view the iterations in the negotiation by clicking “session evolution” tab

· Video clip: Viewing the results in the Joint Gains, 0:59 mins

o with sound (avi 836 kb)

o no sound (avi 188 kb)

o animation (gif 181 kb)

· Analyse the results of the sessions and answer the questions at steps 7 and 8.

7. Questions about the negotiation process

· Present the negotiation processes graphically on the (time, price)-plane. To help to do this you can use the MS excel sheet template.

· Does the negotiation converge?

· Does the initial tentative agreement affect the result? Why?

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8. Summary questions

· Do you think that the final agreements are Pareto optimal?

· Do the tentative agreements seem jointly improving?

· Would you be willing to accept one of the final agreements? Why?

· Do you think that you could use this approach in some real problem of your own? Which?

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