HE3001 Tutorial 1 Solutions

1. Consider the following game matrix.

Player B

Left Right
Player A Top a, b c, d
Bottom e, f g, h

(a) If top and left are strictly dominant strategies, then what do we know the relationship of the
parameters?
From Player A’s strictly dominant strategy
a>e
c>g
From Player B’s strictly dominant strategy
b>d
f>h

(b) If (top, left) is a Nash equilibrium, then what do we know the relationship of the parameters?
a >= e (Best response of Player A)
b >= d (Best response of Player B)

(c) If top and left are strictly dominant strategies, will (top, left) be a Nash equilibrium? Why?
Yes. Strategies which are dominant will always form a Nash equilibrium because they are the
best responses for any action.

(d) If (top, left) is a Nash equilibrium, must the strategies be strictly dominant strategies? Why?
No. A nash equilibrium only requires each player’s strategy to be a best response to the
other’s (fixed) strategy. It doesn’t require the strategy to be a best response to all strategies
of the other player.

The conclusions of parts c) and d) can be seem by the conditions that we derived in parts a) and
b).
2. Consider the following game matrices.

For this question, remember that at each step, we are randomly selecting a player, and
attempting to eliminate a single dominated strategy if possible. This means that there can be
different orders of elimination as long as at each step, we eliminate one of the player’s
dominated strategies.

(a) Is the game strictly dominance solvable? Try this starting from (i) Player A and (ii) Player B.
Player B

L C R
U 3, 0 0, -5 0, -4
Player A M 1, -1 3, 3 -2, 4
D 2, 4 4, 1 -1, 8

Yes.
(i)
M is strictly dominated by D for Player A.
Given the remaining entries, C is strictly dominated by L (and R) for Player B.
Given the remaining entries, D is strictly dominated by U for Player A.
Given the remaining entries, R is strictly dominated by L for Player B.

The remaining outcome is (U,L)

(ii)
V1:
C is strictly dominated by R for Player B.
Given the remaining entries, M is strictly dominated by U for Player A.
Given the remaining entries, we cannot eliminate anything for Player B.
Given the remaining entries, D is strictly dominated by U for Player A.
Given the remaining entries, R strictly dominated by either L for Player B.
The remaining outcome is (U,L)

V2:
C is strictly dominated by R for Player B.
Given the remaining entries, D is strictly dominated by U for Player A.
Given the remaining entries, we cannot eliminate anything for Player B.
Given the remaining entries, M is strictly dominated by U for Player A.
Given the remaining entries, R strictly dominated by either L for Player B.
The remaining outcome is (U,L)
This example shows that when an outcome is obtained via elimination of strictly dominated
strategies, it will be unique, no matter the order, it is also the unique Nash equilibrium.

(b) Is the game weakly dominance solvable? Try this starting from (i) Player A and (ii) Player B.

Player B

L C R
U 6, 0 3, 1 3, 2
Player A M 5, 3 5, 2 3, 2
D 6, 2 4, 2 4, 2

Yes, it is dominance solvable as we can arrive at one of the entries in the table through iterative
elimination of weakly dominant strategies.

(i)
V1
U is weakly dominated by D for Player A.
Given the remaining entries, R is weakly dominated by L for Player B.
Given the remaining entries, cannot eliminate anything for Player A.
Given the remaining entries, C is weakly dominated by L for Player B.
Given the remaining entries, M is weakly and strictly dominated by D for Player A.

The final outcome is (D,L).

(ii)
V2
C is weakly dominated by R for Player B.
Given the remaining entries, M is weakly (strictly) dominated by U (D) for Player A.
Given the remaining entries, L is weakly dominated by R for Player B.
Given the remaining entries, U is weakly and strictly dominated by D for Player A.

The final outcome is (D,R).

V3
C is weakly dominated by R for Player B.
Given the remaining entries, U is weakly dominated by D for Player A.
Given the remaining entries, R is weakly dominated by L for Player B.
Given the remaining entries, M is weakly and strictly dominated by D for Player A.

The final outcome is (D,L).

V4
C is weakly dominated by R for Player B.
Given the remaining entries, U is weakly dominated by D for Player A.
Given the remaining entries, M is weakly dominated by D for Player A.
No further eliminations can be made for Player B because he is indifferent: there is no final
outcome for this sequence of eliminations.

Notice that there are multiple outcomes depending on what is eliminated (V2 and V3). Here it is
because L is sometimes weakly dominated by R, and in other cases R is weakly dominated by L.
(This can only occur if there is indifference in some entries when eliminating.) In some cases, a
sequence of eliminations might not lead to single outcome (V4).

Note that despite the last point, the game is still weakly dominance solvable because in at least
one sequence of eliminations of weakly dominated strategy, we reach a single outcome.

It can also be checked that these are all pure strategy Nash equilibria as well.

(c) In a two-player game, when both players have strictly dominant strategies, will the Nash
equilibrium be unique? What about if both players only have weakly dominant strategies?

These properties in a) and b) hold more generally:

If both players have strictly dominant strategies these will also form the unique Nash equilibrium
(each player only has 1 strategy which is the best response all the time).
If both players only have weakly dominant strategies, then there might not be a unique Nash
equilibrium. For example, below, there is a weakly dominant strategy for each: Up and Left, but
there are two NE: (U,L) and (D,R).

Note the bottom (D,R) although being a NE, does not survive elimination of weakly dominated
strategies.

3. In a two-player voluntary public goods game, each player is given $10 and can choose
between two options: keeping the money or contributing it to a “public fund”. Money in the
public fund gets multiplied by 1.6 and is then divided equally between the two players. The
outcomes are as follows:
 If both contribute their $10, each gets back $16 from the Public fund ($20 × 1.6/2 =
$16).
 If one contributes and the other does not, the contributor gets $8, and the non-
 contributor receives $18 (original $10 + $8 from the public fund).
 If neither contributes, both have their original $10.

The payoff matrix for this game is:

Player B

Contribute Keep
Player A Contribute 16, 16 8, 18
Keep 18, 8 10, 10

(a) Does this game have a dominant strategy equilibrium?

Yes. Both keeping their money is a dominant strategy equilibrium.

Let us consider a more general version of the voluntary public goods game with N players.
Each player can contribute either $10 or nothing to the public fund. All money that is
contributed to the public fund gets multiplied by some number β >1 and then divided
equally among all players in the game (including those who do not contribute.)
(b) If β >1, which of the following outcomes gives the higher payoff to each player? 1) All players
contribute their $10 or 2) all players keep their $10.
All players contribute their $10. This tells us that it is efficient for everyone to contribute.

(c) Suppose that exactly K of the other players contribute. If you keep your $10, you will have
this $10 plus your share of the public fund contributed by others. What will your payoff be in
this case? If you contribute your $10, what will be your payoff?

If keep money: 10+10

βK
N
β (K +1)
If contribute: 10
N

(d) If β=3 and N=5 , what is the dominant strategy equilibrium for this game?
All keep their $10.
If K other players contribute, a player’s payoff will be 10+30K/5 if he keeps the $10 and 30(K
+ 1)/5 if he contributes it. 10+30K/5 - 30(K + 1)/5 = 4 > 0, so we see that for all possible K, the
payoff from keeping the money is always higher.

(e) In general, what relationship between B and N must hold for “Keep” to be a dominant
strategy?
We ask when the payoff from keeping is larger than the payoff from contributing. Using the
βK β (K +1) β
formulas in (c), we want 10+10 >10 , which reduces to <1 .
N N N
 β . This can only happen if
Conversely, for contribute to be a dominant strategy, we need >1
N
contributing is efficient ( β >1 ¿ and it is also high enough such that the free rider effect does
not dominate.

4. Evangeline and Gabriel want to meet each other. There are two possible strategies available
for each of them. These are Go to the Party or Stay Home and Study. They will surely meet if
they both go to the party, and they will surely not otherwise. The outcomes are as follows:
 If both go to the party, the payoff to meeting is 1,000 for each of them.
 If one of them goes to the party while the other stays at home, the one who goes to the
party gets a payoff of 1, while the one who stays at home gets 0.
 If both stay home, each get 1.

The payoffs are described by the matrix below.

Gabriel

Go to Party Stay Home

Evangeline Go to Party 1000, 1000 1, 0
Stay Home 0, 1 1, 1

(a) Is there any outcome in this game where both players are using weakly dominant strategies?
(Go to Party, Go to Party)

(b) Find all of the pure-strategy Nash equilibria for this game.
(Go to Party, Go to Party) and (Stay, Stay).
If a players believes that the other will go to the party, he or she will also go; however, if
players believes that the other would stay, he/she will stay.

(c) Which Nash equilibria do you think is more reasonable and likely to emerge?
(Go to Party, Go to Party).
As you will see in the next tutorial, their best response will depend on the belief that the
other is also going to the party. Since the payoffs for going to the party are much higher than
both of them staying at home (payoff dominant), just a small belief that the other is going to
be party will convince them to go as well. This would makes it more likely that the go to party
outcome will occur.

Conceptual Questions

5. Construct a payoff matrix with two players where:

(a) Player 1 has a dominant strategy.
 (b) Player 2 has a dominated strategy but no dominant strategy.

Player 2
left central right
left 3,2 2,1 1,1
Player 1 central 5,0 4,0 2,1
right 2,0 4,3 1,4

 Player 1 has a dominant strategy of choosing "central."

 Player 2 has a dominated strategy of choosing "central," as they always get a higher
payoff by choosing either "left" or "right," regardless of Player 1's choice.

Please note that the numbers in the matrix are just examples, and any logical solution is correct.

6. Is there always a (pure-strategy) Nash equilibria in the following cases:

(If no, construct an example to show why)
(a) Players 1 and 2 have a strictly dominant pure strategy. Yes
(b) Players 1 and 2 have a strictly dominated pure strategy. No, add on a dominated strategy
for each player to the matching pennies game.
(c) Player 1 has a strictly dominant pure strategy. Yes, Player 1 will only play this strategy in
equilibrium. Player 2 must have a (weak) best response to this, which will be the nash
equilibrium.