Many player rendezvous search:What to do after partial meetingStick together or split and meet

Lyn ThomasCORMSIS

University of Southampton UK

Rendezvous Search• N players placed according to a known distribution in a known region wish

to meet all together in minimum expected time. • Region is discrete set of N points ( represented by a graph ) and meet

together means on same vertex, or region is compact set in which case there is a detection radius d and all are within the detection radius.

• Points of region are homogeneous so no focal point ( a la Schelling) apart from geometry

• M players can move and S are stationary• Moving players are either indistinguishable so all must use the same

strategy - symmetric game• or moving players are distinguishable so can use different strategies –

asymmetric game ( Alpern )• What happens if some but not all meet?

– Stick together– Split and meet– Combination of these

Examples

• Looking for children who are lost : parents move –kids stationary or move

• Friends meeting at social event, new place• Search and rescue• Distress frequencies• Family looking for new jobs+ accommodation + schools

– Stick is sign agreement for that job/house– Chance it will disappear if move to others

• Parallel computing– Processor passes on info ( stick) or keep processing and pass

updated info in future(move)

Finding your kids when they are lost (JORS 43, 637 (92)

• M=2, S=1, circle radius R together with centre point

• If mover finds stationary kid then stick together• Movers agree to meet in centre K times where

time between i-1th and ith meeting is ai

• Other authors solved it for other geometries

2 2( 1) ( 1)

1( ) ( 1 2 )

2i

RK K K K

Ra K i

K

M mover –rendezvous game• Initially each player randomly placed on N of the vertices. (Also deal with players

placed independently )• No meetings, rules exchanges or any other form of communication between the

players before game starts. (So symmetric version of rendezvous game). • At each turn, each player decides to stay or move to some other vertex• Two or more players at same vertex can communicate and agree joint strategy

for rest of game. • • General Symmetric Strategy (sym ) in N-rendezvous games• When players meet they relate where they have been, whom they met, all

agreement for future meetings. So players who meet have identical information. • Subsets of players (randomly chosen) agree to meet at vertex all know how to get

to at some future time. • Different subsets can agree to meet at different times in different places . • Randomly chosen subsets ( could be different groupings) can decide to have a joint

next move. • Strategy is a mixture of randomly going to one of the vertices they do not know

the way to and some possibly vertex dependent probability of going to vertices that the subset of players know the way to.

•

Symmetric and Symmetric stationary no-memory strategies

• Let TM( ), sym be the expected number of moves until all M players rendezvous, under policy .

• is minimum expected moves until rendezvous starting with players at different vertices.

• • sym

nomem is set of symmetric no-memory strategies• Since players with no memory cannot remember the past so no point in arranging

meetings. So randomly chosen subset of players at same vertex have probability of staying where they are or moving randomly.

• The optimal no-memory stationary symmetric strategy is where players stick together once they meet.

• The optimal no-memory stationary symmetric strategy for N vertex complete graph has probability (1/ N) of staying at current vertex and so randomly goes to all vertices.

inf ( )symM MT T

inf ( )symnomem

M MS T

General Results for stick together strategy• Stirling number of the second kind S(k, r) is number of ways of choosing r separate

groups from k people.• Defining [N]r = N(N-1)(N-2)...(N-r+1) then xk = r:0rk S(k,r)[x]r .So S(0,0)=1 , S(n,0) = 0,

S(k+1,r) = S(k, r-1) + rS(k,r)• Conditioning on the first move in the k “player” case leads to the formula•

1

11 ( , )[ ]

k

k r rkr

S S k r N SN

M N 2 3 4 5 6 7 8 9 10 50 100

1 SM 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

2 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 50.00 100.00

3 3.86 5.20 6.54 7.88 9.21 10.55 11.88 13.21 66.55 133.22

4 5.78 7.28 8.79 10.29 11.79 13.29 14.80 74.80 149.80

5 7.71 9.32 10.92 12.52 14.12 15.73 79.74 159.74

6 9.66 11.33 13.00 14.67 16.33 83.01 166.35

7 11.61 13.33 15.04 16.76 85.35 171.06

8 13.57 15.32 17.07 87.09 174.59

9 15.53 17.31 88.44 177.33

10 17.50 89.52 179.52

To compare stick and split and meet use simulation• How players arrange meetings in ‘split and meet’ strategy in simulation• 1. If none of the players has a meeting scheduled, meeting is arranged at same

location in t turns ahead.• 2. If more than one meeting already arranged when players exchange information,

then everyone invited to earliest of meetings as well as one arranged t moves ahead at current location.

• Result of simulation on 50 vertex graph