$1.$ Territoriality

Now, we wish to return to the Hawk$-$Dove model but to more closely tie this model to animal territoriality by introducing $$signals$$ about which player arrived first. We$$ll allow one player to arrive before the other, each player to get a signal of this, and allow players to condition their choice of action on their signal. We formalize as the following Bayesian Game:

A state of the world, $\backslash$omega is randomly chosen from the set $\backslash$Omega $=\{1,2\},$ where $1$ represents the case where player $1$ arrived first, and $2$ the case where player $2$ arrived first. We will assume each state is equally likely. Players do not directly observe the state. Instead, each player, i receives a private signal si that is independently drawn from the set S $=\{1,2\}.$ Let si $=\backslash$omega with probability $1\backslash$epsi $,$ for some $\backslash$epsi in $[0,21).$ We can interpret $\backslash$epsi as the amount of noise in players$$ signals, and $\backslash$epsi $=0$ as the case where both players know, with certainty, who arrived first.

After a state, $\backslash$omega $,$ and signals, s$1,$s$2,$ are drawn, players play the standard Hawk$-$Dove game and can condition their action on their signal. Recall, the standard Hawk$-$Dove game has the following payoffs matrix:

Hawk Hawk $"$ v$2$c$,$v$2$

Dove v$,0$

Dove

v$,0$ # v

$2$

Consider the Strategy profile where each player plays H if and only if they got the signal that they arrived first. I.e$.(\backslash$sigma $1,\backslash$sigma $2),$ where $\backslash$sigma i$($si $=$ i$)=$ Hawk and $\backslash$sigma i$($si $=$ i$)=$ Dove. We will call such a strategy the $$Territorial Strategy$.$

$($a$)$ Start by assuming players know with certainty who arrived first, i$.$e$.\backslash$epsi $=0.$ Show that it is a Bayesian Nash Equilibrium, for both to play the Territorial Strategy.Territoriality

Now, we wish to return to the Hawk$-$Dove model but to more closely tie this model to animal

territoriality by introducing 'signals' about which player arrived first. We'll allow one player

to arrive before the other, each player to get a signal of this, and allow players to condition

their choice of action on their signal. We formalize as the following Bayesian Game:

A state of the world, $$ is randomly chosen from the set $=\{1,2\},$ where $1$ represents the

case where player $1$ arrived first, and $2$ the case where player $2$ arrived first. We will assume

each state is equally likely. Players do not directly observe the state. Instead, each player, $i$

receives a private signal $s_i$ that is independently drawn from the set $S=\{1,2\}.$ Let $s_i=$

with probability $1-lon,$ for some $lonin[0,\frac{1}{2}).$ We can interpret $lon$ as the amount of noise in

players' signals, and $lon=0$ as the case where both players know, with certainty, who arrived

first.

After a state, $,$ and signals, $s_1,s_2,$ are drawn, players play the standard Hawk$-$Dove game

and can condition their action on their signal. Recall, the standard Hawk$-$Dove game has the

following payoffs matrix:

Consider the Strategy profile where each player plays $H$ if and only if they got the signal that

they arrived first. I.e$.({}_1,{}_2),$ where ${}_i(s_i=i)=$Hawk and ${}_i(s_{i}i)=$ Dove. We will call

such a strategy the 'Territorial Strategy'.

$($a$)$ Start by assuming players know with certainty who arrived first, i$.$e$.lon=0.$ Show that

it is a Bayesian Nash Equilibrium, for both to play the Territorial Strategy.

$($b$)$ Next consider an arbitrary value of $lonin(0,\frac{1}{2}).$ Find conditions on $c$ and $v$ under which

it is a Bayesian Nash equilibrium for both to play the Territorial Strategy.