jai4

Theoretical approach to the sitting problem

Framework

We define an abstract object to represent a lecture hall:

L:= < N,K,W >

Where:

N is the number of seats in a row

K is the number of rows in the hall

W is the effective width of the screen (or lecturer)

Assumptions:

Each unit square both contains a desk and is an avenue of travel
Search is global, each agent sees the whole matrix
Agents enter sequentially

Model 1

Assumptions:

Agents are singletons
Agents have identical preferences

The agents choose the best seat, attempting to maximize their utility function that is given by:

U_i = -( a t + (1-a )b )
Where:
t = x+y
b = K-y

When agents are more concerned with minimizing their distance from the screen (or lecturer), a would be relatively small.

a = 0

a < ½

We can see that in this regime the agents bunch near the front, which is not surprising.

a = ½

a =1

Starting with a simple model, which we could solve analytically, we are thus able to derive solutions for some extreme cases. We can extend this particular model by introducing heterogeneous agent population.

Extension:

If we imagine that the agents differ according to a values (e.g. a = Þ "Psyched" a = 1Þ "slackers")

We find:

Results robust to different functional forms (?), Realistic (?)

Model 2

Say people enter in a sequence of small groups, which want to sit in certain arrangements:

Assumptions:

Say a is small

Agents can not pass through occupied squares

Agents can move one square per time step, or wait or sit down

Results:

If agents groups are identical, seating allocation is straightforward.

If agent group differ, 3 effects:

The piece below a group may be indifferent between locations over which the piece above has strong preferences. This implies that the piece above has incentive to wait.

The piece above may "overtake", limiting incentive to wait.

Faster entry rate of groups increases potential for strategic interaction.
Conjecture: Strategic interaction may result in efficiency.

Iris Ginzburg
Asim Khwaja
Justin Smith

16 June 1998