Game Theory

Game theory is the process of mathematically modelling human behaviour and predicting what actions players might take.

Assumptions:

All players are rational i.e. attempt to maximise their own objectives All players are intelligent i.e. knows everything about the game such as their own and rival

players preferences and the consequences of each players actions Each player knows all available actions (A) Each player knows all available payoffs (V) where V=f(A) Each player knows the level of utility they will get from each payoff where U=f(V(A)) Each player prefers more utility to less such that if U (V) > U (V’) then V (A) is he preferred

payoff/action.

It is an important assumption that all players are rational as if player behaviour was stochastic then it would be impossible to draw meaningful conclusions form our analysis.

Criticisms of standard game theory:

Simplicity – in the real world games are incredible complex and it is almost impossible to correctly identify all players involved and assume they all have well defined preferences and are aware of all available actions/payoffs (we assume players are intelligent, doesn’t hold in the real world.)

Rationality – as mentioned before we assume all players are rational for the purpose of analysis, if behaviour is stochastic then conclusions are meaningless

Battle of the sexes game

Pl.2F O

Pl.1 F 2, 1 0,0

O 0,0 1,2

Here we have a classic “battle of the sexes game”, this game consists of n = 2 players, player 1 is a male and player 2 is female. This is a Non-sequential game which means that both players move at the same time (this is an important feature of the game, if the game were sequential then the outcome would be different).

Both players choose to go to the opera (O) or the football (F). Player 1 (the male) would prefer to go to the football, player 2 (the female) would prefer to go to the opera, the payoffs for each action is given in the Payoff Matrix. Note that both players get 0 if they don’t do the activity together, as both player would prefer to do the others preference than do theirs on their own.

Strategies – these are the actions that both players could take. What is the strategy set for each player?

Pl.1 → S1,1 = F S1,2 = O

Pl.2 → S21 = F S2.2 = O

I.e. player 1 can play football or opera as can player 2.

What are the outcomes of this game and the associated payoffs?

(F,F) = 2,1 (F,O) = 0,0

(O,F) = 0,0 (O,O) = 1,2

So how will the game play out?

Well in its current 1-shot, non-sequential form it is impossible to predict how the game will play out. In this 1-shot, non-sequential model there is no dominant strategies or Nash equilibrium. I.e. if player 1 plays football then player 2 should play football also as this will give them the best payoff (1>0) BUT if player 1 plays O then player 2 should also play O (2>0).

The problem here is that both players want to do their activity together but neither can observe what the other plays. If the game was Sequential i.e. one player moves and then the next moves then player 2 would observe what player 1 plays and choose the same action as them. If the game were dynamic i.e. played out over n periods then both players would get a feel for what the other is likely to play, i.e. assume that player 1 plays F for 2 periods, then by the 3rd period, player 2 is knows player 1 is likely to play F again so player 2 will play F also.

A prediction for this game can only be made when the game is either Dynamic or sequential but not in its non-sequential, 1-shot form, what pans out is anyone’s guess.

Dominant and Weakly dominant strategy

Each player has a strategy set/profile, i.e. an action or a combination of actions that they will play through the game. In the Non-sequential, 1 shot-battle of the sexes there was no clearly defined strategy that each player should play. However in some non-sequential games, there may be a dominant or strictly dominant strategy.

Strictly dominant strategy – If player i plays a strictly dominant strategy then the payoff for i us always largest regardless of what player j plays. Example: Prisoners dilemma

Pl.j

S C

Pl.i S 2, 2 5, 1

C 1, 5 4, 4

Here we have the classic prisoners dilemma, again this is a 1 shot, non-sequential game. Both players move at the same time and neither can observe the others actions. Both players have complete knowledge i.e. both players know the payoffs associated with each action for both themselves and the other player. We also assume players are rational and have a negative preference towards jail i.e. the objective of both players is to minimise the payoff.

The game is as follows; players i and j have been caught doing a crime and both now face prison sentences, the payoff matrix represents length of jail sentence in years. Both players are taken away for questioning and must decide whether to confess or stay silent. If both are silent they get done for a lesser crime, if both confess both receive the maximum sentence minus 1 year for their co-

operation, if one confess and one stays silent, then the one that confesses gets off with a light sentence whereas the player that stayed silent get the maximum sentence.

The strategy set for both players it:

Si,1 = S Si, 2 = C

SJ, 1= S SJ, 2 = C

That is player i can be silent (S) or confess (C), as can player j. Is there a dominant strategy to either player?

The answer is yes. If player i stays silent (S) then player j should confess as then player j would only get 1 year in prison (1≻2). If player i confess (C) then player j should also confess to avoid the maximum prison sentence (4≻5). Note that this game is symmetric in that both players face the same strategy sets, payoffs, have the same information about the game and move at the same time. As such we know by symmetry that if a dominant strategy exists for player j then it is also true that the same strategy is a dominant strategy for player i.

As shown confess is a dominant strategy for both i and j, so if both players are playing rationally then both players confess, the game goes to the Nash equilibrium at (C, C) = 4, 4. Both players get 4 years in prison. Note we have assumed that neither player takes into account the others utility when making their decision i.e. its every man for himself.

If a dominant strategy exists for any player then it is assumed that as long as that player is rational then they will play the dominant strategy for as many periods as the strategy remains dominant.

Not Pareto optimal

What is interesting from the prisoner’s dilemma is that it is no Pareto optimal from the point of view of the players. A game is at a Pareto optimal point of neither player can be made better off without making the other worse off. Well let’s look at all the other outcome and compare it to the Nash equilibrium of (C, C) = 4, 4.

(C, S) = 5, 1 – well this isn’t Pareto optimal because even though player 2 would be better off as they would get 1 as oppose to 4, player 1 would be worse off, getting 5 instead of 4. Hence not Pareto optimal as player 1 is made worse, the reverse can be said for (S, C) = 1, 5.

(S, S) = 2, 2 – This solution is Pareto optimal, that’s because both players would be made better off by being at this point.

What’s interesting about the Prisoners dilemma

There’s a number of interesting features of the prisoner’s dilemma game. Firstly the Nash equilibrium is not Pareto optimal, i.e. both players could be better off, but when the game is played rationally, this outcome is not achieved.

Furthermore, even if this game was sequential i.e. one player moves and the other player observes that action then Pareto optimality still won’t be achieved. That’s because confess is still a dominant strategy in this set up, say i plays silent in the hope that j will also play silent, well j would choose to confess and get 1 year as oppose to 2.

The only time Pareto optimality will be reached under this format is if both players act irrationally and cares about the others sentence, then they may agree to both stay silent. However these conditions are far beyond the scope of this model and in any game with irrational players, accurate predictions are impossible as their behaviour is stochastic.

Weakly dominant strategy

Weakly dominant strategy – this is a strategy which either makes the player better off or at the very least not worse off than the other player, depending on the other players actions. Example: Golden balls split or steal.

Pl.2

Sp St

Pl.1 Sp 5, 5 0, 10

St 10, 0 0, 0

This is the classic game of Golden balls. There is a jackpot of 10 which both players are playing for; the strategy set for both players is split (Sp) or steal (St). The game is non-sequential, it is a 1 shot game and both players have complete knowledge of the payoffs and strategies available to the other player. Their objective is to maximise their payoff.

Is there a dominate strategy for either player?

No, and that’s a good thing as it would make for a terrible boring T.V show. From the point of view of player 2, if player 1 chooses to split then player 2 should choose to steal as they will get 10 where 10≻5. If player 1 chooses to steal then player 2 is indifferent between the two actions, he will get 0 regardless. As the game is symmetric the same can be said for player 1, he should play steal if 2 plays split and is indifferent if 2 plays steal.

There exists 3 Nash equilibrium to this game (underline both payoffs when the players are indifferent between actions) the only outcome which is not a Nash equilibrium is (Sp, Sp) = 5, 5 which happens to be the point of Pareto optimality.

The interesting thing about this game is that even though there exists 3 Nash equilibria, we can make a pretty accurate prediction as to what will be played out. Note that Steal was a weakly dominant strategy for both players i.e. even though they get 0 when the other steals, they will get the maximum amount if the other chooses to split. Therefore if both players play rationally and neither cares about the others utility then both should choose to steal and both will get 0. Only when 1 or both players play irrationally is any money won.

NOTE THAT IF A DOMINANT STRATEGY EXISTS THEN THIS WILL ALWAYS BE PLAYED, IF NO DOMINANT STRATEGY EXISTS THEN A WEAKLY DOMINANT STRATEGY SHOULD BE EXPECTED TO BE PLAYED. IF NEITHER A DOMINANT OR WEAKLY DOMINANT STRATEGY EXISTS THEN AN ACCURATE PREDICTION FOR A NON-SEQUENTIAL, ONE-SHOT GAME CAN NOT BE MADE.

Methods of solving a game – Iterated elimination of strictly dominated strategies (IESDS)

As we have shown, rational players will never play a dominated strategy; as such one (slightly crude) way of solving a game is to eliminate any strategy which is dominated.

Pl.2

L C R

Pl.1 U 4, 3 5, 1 6, 2

M 2, 1 8, 4 3, 6

D 3, 0 9, 6 2, 8

Consider the following game, both players have different strategies and payoffs associated with those strategies, dependent on what the other plays. What we need to do now if we are to solve this game using IESDS is to look for any dominated strategies and re-move them from the game. What do we mean by a dominated strategy?

Dominated strategy – Any strategy in which it would not be optimal for that player ever to play it.

From the point of view of Pl.1:

If Pl.2 plays L then Pl.1 should play U If 2 plays C then 1 should play D If 2 plays R then 1 should play U

Note that player 1 should never play M, hence this is a dominated strategy.

From the point of view of Pl.2:

If Pl.1 plays U then 2 should play L If Pl.1 plays M then 2 should play R If Pl.1 plays D then 2 should play R

Note therefore C is a dominated strategy, 2 should never plays C.

Removing strategy M and C gives the following payoff matrix:

Pl.2

L R

Pl.1 U 4, 3 6, 2

D 3, 0 2, 8

We now repeat the process with the new payoff matrix.

From the point of view of 1:

If 2 plays L then 1 should play U If 2 plays R then 1 should play U

Hence D is a dominated strategy and should be removed.

Pl.2

L R

Pl.1 U 4, 3 6, 2

Pl.1 has now removed all dominated strategies and now has a strictly dominant strategy to play U. Pl.2 has complete knowledge and knows 1 will always play U. 2 Therefore plays L to maximise his payoff given he knows 1 will play U.

Our Nash equilibrium and predicted outcome is therefore (U, L) = 4, 3 in this non-sequential, 1-shot game.

(For more examples of IESDS see the workshop answers and lecture examples).

Nash Equilibrium

A Nash equilibrium is defined at a point or solution at which no player has any incentive to deviate or change strategies. In our prisoners dilemma from earlier there existed a single Nash equilibrium where both players chose to follow their strictly dominant strategy and confess. At this point both players were playing their strictly dominant strategy and had no incentive to deviate from the Nash equilibrium.

Note that in some games there may exist more than one Nash equilibrium for example consider again the battle of the sexes game.

Pl.2

O F

Pl.1 O 1, 2 0, 0

F 0, 0 2, 1

How does one go about finding the Nash equilibria? Well one must consider, what is the best strategy from both players perspective given the n strategies of the opposing player?

From player 1’s perspective:

If Pl.2 chooses the opera then player 1 should also chose the opera as the objective is to maximise the payoff. (Underline this payoff)

If Pl.2 chooses the football then player 1 should also choose the football. (underline this payoff)

From player 2’s perspective:

If Pl.1 chooses the opera then 2 should chose the opera also, if 1 chooses football then 2 should chose football. (Underline both of these payoffs for Pl.2

Note we now have to strategy profiles or outcomes which have two payoffs underlined. If both the payoffs are underlined then this is Nash equilibrium. I.e. at that point both players are playing their optimum strategy given the strategy played by the opposing player. Neither player has an incentive to change strategy if they are doing the act together i.e. if they are both at the football (F, F) then there is no incentive for player 2 to decide to instead go to the opera (F,O) as this would make both players worse off.

Note that again if this game is non-sequential then which Nash equilibrium the game converges to is impossible to determine without introducing dynamics and mixed strategies.

Pure strategy: A pure strategy is a complete definition of how a player will play the game, for example in the battle of the sexes player 1 might say well i prefer the football and i can’t observe what player 2 will play so i will always choose football as my strategy and hope player 2 also plays football.

Mixed Strategy: This is where the player assigns a probability to each pure strategy and then randomly determines which strategy to play, for example if i am player 1 in the battle of the sexes i may say, “ok well i think 80% of the time player 2 will choose opera” so i will choose assign an 0.8 probability to opera and a 0.2 probability to football and choose at random.

Note also that Nash equilibrium are not always Pareto Optimal, we found earlier in the prisoners dilemma, the Nash equilibrium was where both players confess but the point of Pareto optimality was where both players stayed silent, this is why the outcome of games can appear strange from a 3rd person perspective.

Unique Nash Equilibrium – Not all games have a unique Nash equilibrium, but if a game does possess such equilibrium then we can make a strong prediction about that game. A unique Nash equilibrium is a game with only 1 Nash equilibrium or at least 1 true Nash equilibrium. For example the prisoner’s dilemma has just 1 NE, i.e. both chose to confess and as such we can make a fairly strong prediction that both will confess id the game is played rationally. However the Battle of the sexes has to NE, (O, O) and (F, F) which one the game moves towards is unknown and as such a strong prediction as to the outcome of the game cannot be made. In the case of Golden balls it could be argued that if both players play rationally then both should chose to steal and as such there should exist only 1 NE even though we found there to be 3 where (Sp, Sp) was the only strategy set which was not a NE.

Subgame perfect NE – A Subgame perfect NE exists only in sequential games. This is because in sequential games where players observe the strategies played by the other, players may threaten each other with irrational moves, as such a Nash equilibrium may exist at a non-rational point.

Nash Equilibrium and Pareto optimality, a further example:

Consider the following game, there are two hunters who go hunting but know nothing of each other prior to the game. The have two strategies, they can hunt stags or they can hunt hares. If they hunt hares they both get a payoff of 3 regardless of whether they both chose to hunt hares or not. If they want to hunt stags then they get a payoff of 5 But the catch is they must co-operate together to hunt a stag due to its higher degree of difficulty to catch. The objective of both players is to maximise their payoff, the payoff matrix is given below:

Pl.2

S H

Pl.1 S 5, 5 0, 3

H 3, 0 3, 3

The Nash equilibria have been found at (S, S) where both hunt stags, and (H, H) where both hunt hares. What is interesting about this game is that only one of the NE points is Pareto optimal, that is at (S, S) = 5, 5 where the payoff for both players is greater. However despite this point being Pareto

optimal it is not certain that the players will choose this point. If players do not trust each other then it is likely that both will turn to hares. After all, regardless of what the other plays you will always get a non-zero payoff hunting hares whereas you may get nothing hunting stags.

This game has often been used to determine how much people in society or in a group trust each other.

Dynamic and sequential games

So far we have looked only at games which last for one period or have looked at games in which both players move at the same time. We now go on to look a dynamic games where by players play strategies over a given number of periods and players change their strategies for a given period in response to the actions of other players.

The classic example of how dynamic, sequential games in real life is that of collusion amongst oligopoly firms. Firms may agree to restrict output to its monopoly level and share the monopoly profits from that. However there is also an incentive for a single firm to cheat for 1 period, increasing its output and getting greater than their share of monopoly profits, by having greater output at the monopoly price (given that price is fixed for 1 period). However the other firms will respond in the next period by increasing their output. If the game was a one-shot game then one might suggest all firms will choose to cheat and collusion will be impossible, however in a game with infinite periods firms might decide to stick to the collusive level of output if the long term benefits outweigh the short term cash inflow.

How does a game being dynamic effect the game?

In a game with dynamics the strategies, and strategy stets change for the players, what is a dominant strategy in period t may not be a dominant strategy in period t+1, it depends on the actions of other players in period t.

The best way to represent a game with dynamics or a sequential game is using a tree, for example consider the battle of the sexes where player 1 goes first and then player 2 goes second.

O (0, 0)

(0, 0)

Features of the tree diagram:

The tree diagram starts off with player 1 or move 1 represented by the circle with a 1 inside. From player 1 stems all the available strategies available to them at that time, we know in the battle of the sexes the first mover has only two available strategies go to the opera or go to the football.

Because of these two strategies the game moves to one of the two possible points, if player 1 chooses opera the game moves to the upper branch, if player 1 chooses football then the game

O

O

F

F

F1

2

2

(1, 2)

(2, 1)

moves to the lower branch. Player two now has to choose opera or football and if the action of 1 was observed then two will choose the maximum payoff relevant to the position of the game, i.e. if 1 chose football and 2 observed this then assuming 2 plays rationally then 2 will also choose football, the relevant payoffs are given at the end nodes (2,1) in this case.

NOTE THE DOTTED LINE AROUND THE 2’S. IF A DOTTED LINE IS PRESENT IN THE DIAGRAM IT MEANS THAT 2 DID NOT OBSERVE WHAT 1 PLAYED, I.E. 2 IS UNAWARE OF WHETHER THE GAME WENT TO THE UPPER OR LOWER BRANCH.

Determining the number of strategies –extensive form

The first step in solving a game with dynamics is to determine all of the strategies available to each player; the number of strategies available to the player depends on:

How many times the player moves in the game The number of actions available to him when he moves.

Let’s look at our battle of the sexes case, how many times does player 1 play? Once right? And how many available actions does he have with his one move? two, O or F.

To find the number of strategies available to each player you multiply the number of plays against their possible actions:

Player 1’s strategies: S1 → 1X2 = 2 → (O, F)

I.e. player 1 plays once and has two actions; one multiplied by two is two so we expect one to have two strategies available. (opera, Football)

Player 2’s strategies: S2 → 1x4 = 4 → (OO, OF, FO, FF)

I.e. player 2 plays once but has 4 possible options or routes for the game to go (look at the number of branches) we therefore expect player 2 to have 4 strategies, given above.

Another example:

Let’s now look at the game represented in figure 2, here player one moves twice and player 2 moves once, what are that strategies available to each player?

Player 2: Player 2 moves once and has only 2 actions available to her at her move, her strategies are therefore 1X2 = 2

S2 (BC, BD)

Player 1’s strategy set is more complicated he moves twice and has two options available to him at each move hence the number of strategies available are 2X2 = 4 → S1 = (A; B→E) (A; B→F) (B; B→E) (B; B→F)

Consider finally this figure, how many strategies are available to each player and what are they?

Player one has 3 options at the first move and two options at the second so his number of strategies is 3X2 = 6, they are (CF, CG, BF, BG, AF, AG) i.e. player one chooses C, B or A at the first play and then if the game continues to a second play then he can only chose F or G.

Player two has 2 options at the westerly node and 3 options at the easterly node, the number of strategies for two is therefore 3X2 = 6. These are (DH, DI, DJ, EH, EI, EJ) i.e. take DH, what does this mean? This means that two plays D if the game goes to the left or H if the game goes to the right, EH, what does this mean? If the game goes to the left player 2 plays E, if it goes to the right player two plays H.

Why is knowing the number of strategies available in a game useful? because when we now write the game from extensive form (tree’s) to Normal form (matrix) we need to know the number of rows and columns. In the above game we saw that both players had 6 available strategies and as such a 6X6 matrix would be made.

This would be the matrix representation for the above game. It is useful to be able to represent an extensive form game as a normal form game as it is easier to identify all Nash equilibria and also every extensive form game has a unique normal form representation.

Sequential battle of the sexes, represented in normal form

As mentioned before in the battle of the sexes player 1 has 2 strategies and player 2 has 4 hence our matrix will be a 2x4.

Pl.2

OO OF FO FF

Pl.1 O 2, 1 2, 1 0, 0 0, 0

F 0, 0 1, 2 0, 0 1, 2

In the first box, player 1 chooses 0, Player 2 cant observe what one has chosen so decides to always choose O regardless of what 1 does, fortunately 1 chose O and hence 1 (who prefers the opera gets a payoff of 2 and 2 (who prefers football gets a payoff of 1). In the box below 2 always go to the opera but 1 chose to go to the football hence both get 0.

In the next column, 2 chooses a mixed strategy where they go to the opera in the first instance but change to football in the second, luckily for him 1 chose to go to the opera first and football second. The third column is again a mixed strategy but the wrong way round leading to 0 payoffs as both go to different events and the final column is a payoff where 2 chooses a pure strategy of always going to football.

Find the Nash equilibrium

You follow the same procedure to find the NE as before, if 1 plays O then 2 is indifferent between OO and OF (underline the payoffs). If 1 plays F then 2 is indifferent between OF and FF (underline these). Now play the columns:

If 2 plays OO then 1 should play O, if 2 Plays OF then 1 should play O, if 2 plays FO then 1 is indifferent between O and F and finally if 2 plays FF then 1 should Play F (Underline all).

Note we now have 3 NE, they are (O, OO) (F, FF) and (O, OF).

How do we solve?

We have found the three NE, however which one do we think the game will converge towards? to do this we must find the Subgame perfect NE. Subgame perfection requires:

Mutual best-response on the equilibrium path Mutual best response off the equilibrium path

To find the SPNE you slit the tree into all the possible sub-games and find the best response in each subgame.

Recall the battle of the Sexes example from earlier, there are 3 subgames within this game. To find the SPNE you must work backwards through the subgames. If first subgame (Upper branch) what is the best response that player 2 could make? Well if the game has gone to the upper branch then 1 has chosen O so 2 should also choose O. In the second subgame (lower branch) if the game has gone here then 1 has chosen F so 2 should also choose F i.e. 2’s optimal strategy is OF.

Therefore our Subgame optimal NE for the battle of the sexes is (O, OF) that is 2 choses O or F depending on what 1 choses, 1 choses O so they both go to the Opera. Why did they go to the opera? because O moved first and prefers the opera, so rationally chooses to go to the opera to maximise her potential payoff, 2 has a strategy in which he will match whatever 1 does hence 2 also goes to the opera. If player 2 went first then he would chose the football and 2 would follow by going to the football also.

Note that we saw earlier that the battle of the sexes can only be solved when one player moves first, the solution to the game is that the player who moves first will chose their preference and the other player will follow. This is the SPNE but note not all NE are sub perfect, we found 3 Nash equilibrium, but only one was sub game perfect.

Questions /examples

Question 1:

1i) Represent the Prisoners dilemma in extensive form given that player 1 chooses whether to confess or stay silent whilst player 2 cannot observe this action. (Use the payoffs from earlier)

1ii) Represent prisoner’s dilemma in Normal form using the extensive form from 1i).

1iii) Find all Nash equilibria and determine whether any are sub-game perfect

1iv) Make a prediction for this sequential version of prisoner’s dilemma and comment on your results

Answers:

i) S

S C (5, 1)

(1, 5)

C

C (4, 4)

S1

2

2

(2, 2)

ii)

Player 1’s strategies: S1 (S, C)

Player 2’s strategies: S2 (SS, SC, CS, CC)

Pl.2

SS SC CS CC

Pl.1 S 2, 2 2, 2 5, 1 5, 1

C 1, 5 4, 4 1, 5 4, 4

iii)

Play the rows:

If 1 stay silent then 2 is indifferent between CS and CC, remember the objective is to minimise the payoff in this game, (underline).

If 1 chooses confess then 2 is indifferent between SC and CC.

Play the columns:

If 2 chooses SS, 1 should chose C If 2 choses SC, 1 should chose S. (note that in this strategy 2 will match any action of 1 that’s

why 1 should chose to be silent) If 2 choses CS, 1 should chose C If 2 choses CC, 1 should chose C

The Unique Nash equilibrium is found at (C, CC) where both players chose to confess regardless of what the other player does. The NE is subgame perfect as it yields the lower payoff for player 2 given what player 1 plays, there is never a reason to play S.

iv) As expected the prisoners’ dilemma gives a Unique NE regardless of whether the game is sequential or non-sequential. Both players chose to confess regardless of what the other does, i.e. follow their strictly dominant strategy. The only way in which the result of the game may change is if both players payoff function is adversely affected by the others, i.e. if one cares a lot about the other and doesn’t want them to go to jail then they may stay silent, particularly in a sequential game where one can observe the other.

One might think under observation and given the players care about each other’s payoff that each will agree to stay silent (similar to collusion) and base a trigger strategy on the other person confessing, i.e. i will stay silent unless the other person confesses, in which case i will also confess.

Question 2:

Consider the following game, Player 1 and player 2 each put a penny on the table simultaneously, if the two pennies come up the same side (heads or tails) then player 1 gets both pennies, otherwise player 2 gets the pennies.

2i) If the game is a 1-shot, non-sequential game is there a Nash equilibrium? Is there a dominant strategy? Can a strong prediction as to the outcome of the game be made?

Now assume that the game is sequential and player 2 observes the actions of player 1 and reacts accordingly.

2ii) Represent the game in extensive form

2iii) Represent the game in normal form and find the Nash equilibria, are any subgame perfect?

2iv) Can a strong prediction be made about this game, does either player have an advantage? Who would generate the highest payoff if the game were repeated over n periods?

Answers:

2i)

Pl. 2

H T

Pl.1 H 2, 0 0, 2

T 0, 2 2, 0

Play the rows:

If 1 plays heads then 2 should play tails If 1 plays tails then 2 should play heads

Play the columns:

If 2 plays heads then 1 should play heads If 2 plays tails then 1 should play tails

In this Non-sequential, one shot game there is no Nash equilibrium nor is there a dominant or weakly dominant strategy. A prediction is therefore impossible in its current format as which strategy each player will follow is completely stochastic; both have a 50/50 shot at winning.

2ii)

H (2, 0)

H (0, 2)

H (0, 2)

T

T (2, 0)

2iii)

S1 (H, T)

S2 (HH, HT, TH, TT)

1

2

2

T

Pl.2

HH HT TH TT

Pl.1 H 2, 0 2, 0 0, 2 0, 2

T 0, 2 2, 0 0, 2 2, 0

Play the rows:

If 1 plays H, 2 is indifferent between TH and TT If 1 plays T, 2 is indifferent between HH and TH

Play the columns:

If 2 plays HH 1 plays H If 2 plays HT, 1 is indifferent between H and T If 2 plays TH, 1 is indifferent between H and T If 2 plays TT, 1 plays T

The two Nash equilibria for this sequential game is (H, TH) and (T, TH), (H, TH) is the subgame perfect Nash equilibrium.

iv) A subgame perfect Nash equilibrium exists at (H,TH) essentially this means that player 2 will always play the opposite to what player 1 plays. This is feasible given the game is sequential and 2 observes what 1 plays. Player 2 has second – mover advantage and as such will always win over an infinite number of periods. If player 2 could not observe what 1 played, then there would be no second mover advantage and we would be back to the previous example where no prediction could be made. Player 2 would play a mixed strategy based on what they think 1 is likely to play.

Question 3

Following the Greece bailout programme of 2014/15, which ended in Greece successfully getting an extension on its loans, describe the event in an extensive form game and show why the EU giving Greece and extension was the subgame perfect Nash equilibrium.

Follow (-3, -5)

Accept (5, -1)

Challenge

Extension (-2, 3)

Press

Leave (-5, -3)

Greece

EU

Greece

Let’s look at the actions available to each player:

SG = (F, CE, CL)SEU = (Accept, Press)

Greece is player 1 as it moves first to decide whether to accept the current terms of the loan or challenge to try to agree new ones. The normal form game will therefore be a 3x2 matrix.

E.U

Accept Press

Greece Follow -3, 5 -3, 5

Challenge, extension -2, 3 -5, -3

Challenge, leave -5, -3 -5, -3

Therefore the unique Nash equilibrium for this game was for Greece to challenge, knowing the EU would not accept but then allowing them to ask for an extension, knowing the EU wouldn’t allow them to leave in fear of losing other countries such as Portugal and Ireland. Greece therefore took advantage of the game and made things for themselves marginally better at the expense of the EU.