the easiest way to solve problems with a continuous strategy space, is to solve for players’ best response (BR) function. The BR function tells you, for any t−i that the other player chooses, what choice of ti would maximize player i’s payoffs? I.e. BRi(t−i) = t

(a) Often, the easiest way to solve problems with a continuous strategy space, is to solve
for players’ best response (BR) function. The BR function tells you, for any t−i that
the other player chooses, what choice of ti would maximize player i’s payoffs? I.e.
BRi(t−i) = t
∗
i
, where u (t
∗
i
, t−i) ≥ u(ti
, t−i), ∀ti
. Let’s solve for player 1’s BR function.
i. Suppose t2 < v c . Verify that BR1(t2) = {t1|t1 > t2}.
ii. Next, suppose t2 =
v
c
. What is BR1

v
c

?
iii. Finally, if t2 >
v
c
, what is BR1(t2)?
(b) What is player 2’s BR function?
(c) Now, recall that in a Nash equilibrium, neither player can benefit by deviating. That is,
both players must be playing according to their best response, which can only happen
if t1 ∈ BR1(t2) and t2 ∈ BR2(t1). Find all Nash equilibria in this game. That is, find
all pairs (t
∗
1
, t∗
2
) such that t
∗
1
is a best response to t
∗
2
, and t
∗
2
is a best response to t
∗
1
.

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+µ2
(1−µ)(1−)+µ 
≥
v
2c ≥ max 
1 −
(1−µ)(1−)
2+µ2
(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?

Now, instead, imagine that the imbalance of power causes the costs of fighting to be unequally distributed between the two players (you may go back to assuming c > v/2). In particular, suppose one player incurs a cost of c 8 and the other player incurs a cost of 15c 8 . Write down the payoff matrix of this new game and find all pure Nash equilibria. Does the prediction of the model change this time?

Now, instead, imagine that the imbalance of power causes the costs of fighting to be
unequally distributed between the two players (you may go back to assuming c > v/2).
In particular, suppose one player incurs a cost of c
8
and the other player incurs a cost of
15c
8
. Write down the payoff matrix of this new game and find all pure Nash equilibria.
Does the prediction of the model change this time?

Imagine that there is an imbalance of power between the two players, and that, if both play hawk, player 1 wins with probably q where 0.5 < q < 1. Assume both players still incur the cost c of fighting. Write down the payoff matrix of this new game.

Imagine that there is an imbalance of power between the two players, and that, if both
play hawk, player 1 wins with probably q where 0.5 < q < 1. Assume both players still incur the cost c of fighting. i. Write down the payoff matrix of this new game. ii. Suppose c > v. Find all pure Nash equilibria. Does the prediction of the model
depend on the likelihood one player wins?
iii. Now, suppose v > c > v/2 and find all pure Nash equilibria. Does the prediction
of the model depend on the likelihood one player wins? How do you interpret your
answers to these last two questions?

Two players are contesting a resource worth v > 0. Each can be a dove (passive) or a hawk (aggressive). If one is aggressive and the other passive, the aggressive opponent gets the resource. If both are passive, each obtains the resource with equal likelihood. If both are aggressive, both players incur cost c, and each obtains the resource with equal 1 likelihood. We assume c is large relative to the value of the good, c > v/2. Write down the set of strategies. Then, in the form of a payoff matrix, write the payoffs each player gets from each strategy profile. Finally, find all pure Nash equilibria.

Two players are contesting a resource worth v > 0. Each can be a dove (passive) or a
hawk (aggressive). If one is aggressive and the other passive, the aggressive opponent
gets the resource. If both are passive, each obtains the resource with equal likelihood. If
both are aggressive, both players incur cost c, and each obtains the resource with equal
1
likelihood. We assume c is large relative to the value of the good, c > v/2. Write down
the set of strategies. Then, in the form of a payoff matrix, write the payoffs each player
gets from each strategy profile. Finally, find all pure Nash equilibria.

Write an essay in which you analyze one or more of the following rhetorical devices angela lee duckworth used in her speech, grit: the power of passion and perseverance, to achieve her purpose

Write an essay in which you analyze one or more of the following rhetorical devices angela lee duckworth used in her speech, grit: the power of passion and perseverance, to achieve her purpose. Discussing her experiences as a teacher , referring to findings from research studies, admitting that she doesn’t know exactly how to make students more gritty

explain how the two critics differ in their reading of the primary work. What parts of their argument cannot be made to agree? discuss the way they frame their essays in terms of previous arguments about the work. What are the stakes of these two critics’ claims?

explain how the two critics differ in their reading of the primary work. What parts of their argument cannot be made to agree?
discuss the way they frame their essays in terms of previous arguments about the work. What are the stakes of these two critics’ claims?
evaluate their use of primary evidence. What are the most striking ways they use evidence from Parks’s or Austen’s text? What is most useful or insightful about their readings? Have they left any key passages out? That done, show us what argument you can make in the gap these two critics have left: where is there space for you to make a third argument that continues their conversation? And where would you find support for that argument in the text?