nm0867: okay can i have your attention please we shall do the test in the last 
er fifteen twenty minutes of the lecture and now i will briefly remind you what 
we were doing last week first well and the week before that first we can see 
that er in a game of hawks and doves we found an equilibrium in this game then 
we introduced another strategy of the game bullies and we can see that the 
three subgames of the game of three strategies and that's what we found don't 
er er draw this picture in your notebooks er we'll er draw something like that 
a little later just remind you what they have three strategies if we omit a 
third one then in the remaining two that's what we call the subgame hawk doves 
and equilibriums one-third from hawks er in the subgame consisting of hawks and 
bullies we again had an equilibrium somewhere between those two strategies and 
if we considered and when we considered that such game consisting of doves and 
bullies we didn't find any equilibrium and the 
system was evolving towards bullies er doves did not survive er interaction 
with bullies now the question okay these er well this this and this are some er 
evolutionist table populations for the subgames what happens if we introduce 
the third strategy that's where we start the lecture in other words we found 
three evolutionary stable populations for the subgames this one this one and 
this one the question is are these E-S-Ps stable with respect to introduction 
of the third strategy and that's how we're going to approach it can see that 
for example can see that for example this point er this point is described by 
the vector of i think it's two-thirds two-thirds one-third zero so this is X if 
we multiply that by the pay-off matrix we get the fitnesses and for this case 
it is four-thirds four-thirds and eight-thirds so the hawks and the doves 
they're doing on par which is expected after all it is the equilibrium of the 
subgame but interestingly the bullies are doing 
better than the other two what it means it means that if we introduce a very 
small proportion of bullies to this system it'll drive away from the E-S-P 
because bu-, bullies are doing better than both hawks and doves now it's clear 
what we should do in the general case oh for the other two E-S-Ps the same 
thing describe for example this point or this point by a similar vector 
multiply it by the pay-off matrix and obtain the fitness vector and then i'm 
not really sure what's going on with the lights [laughter] and er just check 
which of the strategies is bigger i skipped the calculations in fact i have 
included them in the problem sheet in the next problem sheet just the 
conclusions so this point is unstable as we found out in the first place er 
this point is stable and this point is unstable so that's what we've got now 
we're going to introduce the notion of a flow chart and this is more or less 
the same kind of picture as we've got here three strategies three coordinates X 
hawks X doves 
and X er X bullies but not any point is allowable in this space clearly we er 
should satisfy the condition that the proportions of the strategies add up to 
one so it's X-H-plus- X-dove- plus- X- bullies add up to unity which cuts and 
of course the the three corners should be positive so this cuts a triangle in 
between the axes and figure one is this one one one and one so what is the flow 
chart we want to show the trajectories of our system on this triangle so what 
what do we need to do we know we need to write down the dynamics equations 
solve them and draw the solutions on the triangle the second and the third 
items on this list on this list er they are effectively er equivalent to 
solving the dynamics equations but we don't really need to be very precise 
about drawing the trajectories we want to er understand just qualitatively how 
the trajectories go and for that we don't really need to solve equations or 
follow these three items 
what we really need to know is is this the equilibrium positions and 
evolutionarist table populations of the subgames and of the game itself this 
particular game doesn't have any E-P any equilibrium positions er in the middle 
of this triangle only subgames have equilibrium positions so then there is two 
two possibilities er at this stage you probably need to stop writing and just 
watch what i'm going to draw here how what i know how trajectories behave on 
the sides of the triangle if i'm here then the system evolves towards bullies 
if i'm here the system evolves towards the equilibrium position on this side 
the same thi-, thing here and here so how what happens if i am somewhere here 
clearly if i'm closer to this side i'll go some way along this side if i'm 
closer to this side then i'll go this way then i go go this way what can happen 
here clearly i can't hit the vor-, er the vertex er because if i am here i will 
go this way so clearly the trajectory has to 
turn at some stage and go this way and here it may go for example toward this 
equilibrium and here this equilibrium two possibilities er like i said one of 
them is this one in this case it can't go this way then it goes this way and 
this case this one can't go this way goes this way the other possibility this 
bit is exactly the same hawks doves bullies equilibrium equilibrium but the 
difference between this would be for this way and in this case which one is 
correct which which of the two scenarios is correct 
sm0868: the first 
nm0867: the first why 
sm0868: because the other two the other two points are unstable 
nm0867: that's correct this implies that this is unstable and this is stable er 
this one is the other way around and we just now we just checked before that 
this is unstable and this is stable so that's the way to go correct wrong and 
that's how we will approach similar problems in the future we won't solve 
differential equations we shall examine er equilibria 
we sh-, well we shall find the equilibria of the system we shall examine the 
stability of those equilibria and then we just draw a flow chart and if we know 
the the flow chart then we'll ba-, we'll basically know everything about our 
system i know that you haven't done differential equations at this stage you 
are going to do that next term i suppose er but i will just introduce bits and 
pieces from the theory of differential equations 
sm0869: excuse me just one question 
nm0867: yeah sure 
sm0869: er you're saying that it's an unstable point for the equilibrium at the 
bottom but is it 'cause it's a subgame of hawks and doves yeah 
nm0867: that's correct 
sm0869: is this a subgame of hawks and doves will it be a stable one 
nm0867: in the subgame yes 
sm0869: subgame but when you 
nm0867: so 
sm0869: introduce the bullies it becomes unstable 
nm0867: yeah that's correct if we're exactly on this side of the triangle we do 
go this way but any trajectory which is infinitesimally above the set will go 
away it will never enter the point so the er separatrix so 
forget this that's an arrow that's what is really going on and the separatrix 
is this bit so to the right from the separatrix the trajectories originate 
basically from er this point and to the left from the separatrix all the 
trajectories originate from this point and well basically at this point you you 
should consider it not as an origin of this trajectory but rather a midpoint of 
this trajectory er so that is er very unfortunate for the doves conclusion 
nm0867: when i was lecturing this course last year er i was curious if it is 
possible at all to modify the rules of the game so the doves will survive and 
it turned out that it is very difficult they can't survive in direction of the 
hawks but if you've got hawks and bullies which act slightly differently with 
respect to doves er doves have very slim chances of surviving still i was 
experimenting and eventually i did find a couple of ways to let er the doves 
survive so the next section is new 
species of oh well mathematically speaking more strategies first retaliators ah 
sorry [laugh] 
sm0870: could you put it back down a little 
nm0867: yeah sure 
nm0867: first i tried to make bullies sillier er remember when two bullies meet 
and fight over a piece of resources they share their win without wasting time 
which gave them three and three and and that's what made them very difficult to 
compete with with respect to doves 'cause er doves do waste time so i tried to 
make bullies waste time and i checked the game consisting of hawks doves and 
silly bullies and it turned out that doves don't survive even with er with this 
modification next i'll try i tried to make doves more clever so the difference 
here is those numbers now doves do not waste time i don't really [laugh] 
remember if this worked but apparently it didn't er because i tried then to 
introduce a species which would protect doves from hawks that is it would 
compete with hawks stronger than it would compete with doves and that's what 
the so-called 
retaliators do basically er the notion of retaliators they beha-, with hawks 
they behave like hawks and with doves they behave like like doves so that is 
you see the difference here er retaliators don't waste time doves do waste time 
and in fact this didn't work either er in a game which includes hawks 
retaliators and doves er er doves do not survive and when er they disappear 
from the system then retaliators don't er differ from hawks and in fact it's 
fairly interesting situation here er if you draw a flow chart for this game 
like hawks er don't write this down just aside hawks do-, er doves and 
retaliators we still have this er unstable equilibrium er but in fact any point 
of this any point of this side is a stable equilibrium if a trajectory from 
here for example hits er a point and the set is going to be stable i don't re-, 
actually remember how the flow chart goes but er i vaguely remember that it is 
like this and every trajectory er ends at this side and the system doesn't 
evolve after that then i can see that well obviously silly retaliators again 
the difference is that silly retaliators waste time unlike their cleverer 
brothers just wait a second and eventually the pinnacle of my creativity 
swingers [laughter] er [laugh] these guys they don't er follow just like 
strategies they follow they behave like hawks or doves with fifty per cent 
probability depending on random things like er what they had eaten the day 
before yesterday or to the the mood they and in fact er i'd [laugh] to tell the 
truth i haven't er examined this situation but er what at-, attracts me in this 
particular thing is that i've met a lot of people like that [laughter] 
sm0871: sorry can you move it down a bit 
sm0872: 
nm0867: s-, sorry say it again 
sm0872: i mean the there are four species now i mean shouldn't you have 
nm0867: no no what i was thinking of is hawks and doves they stay in the system 
in all cases and then i was trying to replace bullies with something else which 
would er 
sm0873: no i th-, i think he means on that table 
sm0872: yeah 
i mean 
sm0874: yeah it's S 
sm0875: should be S 
nm0867: ah yeah that should be S so sorry S-R yeah three strategies when you 
can see the four strategies there er er the system me-, er becomes fairly 
difficult to examine m-, mathematically it's n-, just just cumbersome not 
difficult unfortunately er like i said the numbers here become er fractional er 
i just don't have the patience to examine if how er how this species behaves in 
a in er in a game with other species and i take you understand the meaning of 
those numbers here 
sf0876: 
nm0867: minus-two means both swingers fight choose to fight this means that the 
first well any well the first er swinger runs away er the second gets the prize 
this is the reverse situation the first takes the prize the second one runs 
away and that's they share without waste wasting time er again you can consider 
silly swingers but swingers are again my er the people i know they never waste 
time [laughter] they just do what er er what they want on the spur of the 
moment it would be uncharacteristic of the swinger to but er and i'm afraid 
that's er where we have to proceed with the test