##### We use the Hawk-Dove game to understand animal territoriality, as well as some key attributes of our sense of rights. In particular, when it comes to animal territoriality, we ask, why are ‘fights’ over mates, territory, or food so often symbolic, with one animal typically acquiescing after only a brief engagement? This animal is usually the one that arrived at the resource second. In humans, our sense of rights has a clear analog: we think those who possess (or find or make) an item have a ‘right’ to it, and we are willing to argue or fight in defense of our rights, e.g., when someone takes something that ours. But, we don’t just grab stuff that isn’t ours

Now, to more closely tie the model to territoriality and rights, we’ll allow one player to
arrive before the other, and also allow players to condition their choice of action on their
signal of who arrived first. We formalize this as follows.
A state of the world, ω is randomly chosen from the set Ω = {1, 2}, where 1 represents
the case where player 1 arrived first, and 2 the case where player 2 arrived first. The
probability that player 1 arrived first is µ. 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 = ω with probability 1 − , for some  ∈ [0,
1
2
]. We can interpret
 as the amount of noise in players’ signals, and  = 0 as the familiar case where both
players know who arrived first with certainty.
After a state, ω, and signals, s1, s2, are drawn, players play the Hawk-Dove game and
can condition their action on their signal. Note that the payoffs of the Hawk-Dove game
do not depend on the state or signals.
i. Define the strategy σ

i
as play H iff si = i. Start by assuming players know with
certainty who arrived first,  = 0. Show σ

i
is a Nash equilibrium.
ii. (σ

1
, σ∗
2
) is a (Bayesian) Nash equilibrium if and only if min 
µ(1−)
2+(1−µ)
2
µ(1−)+(1−µ)
,
(1−µ)(1−)
2+µ2
(1−µ)(1−)+µ 

v
2c ≥ max 
1 −
(1−µ)(1−)
2+µ2
(1−µ)(1−)+µ
, 1 −
µ(1−)
2+(1−µ)
2
µ(1−)+(1−µ)

. To show this, answer the following:
A. Suppose player i’s signal indicates she arrived first, si = i. Calculate conditions
such that she prefers not to deviate from σ

i
.
2
B. Suppose player i’s signal indicates she arrived second, si = −i. Calculate conditions such that she prefers not to deviate from σ

i
.
iii. Interpret this condition. In particular, how does  effect whether the proposed
strategy can be an equilibrium? To make things simpler, you can assume players
are equally likely to arrive first, µ =
1
2
.
iv. Suppose that player 1 has complete information: she can observe ω, s1 and s2.
However, player 2 can only observe her signal, s2. If player 2 plays according to σ

2
,
what is player 1’s best response? Does this depend on ω? s1? s2?
v. Now suppose that, with probably q, player 1 observes ω, s1, and s2, and with probability 1 − q, player 1 observes only s1. Player 2 can still only observe her signal,
s2. If player 2 plays according to σ

2
, what is player 1’s best response? Is it a Nash
Equilibrium for player 1 to play according to this strategy and for player 2 to play
according to σ

2
, or would player 2 benefit by deviating?