Vanessa Jaya
Hertford College
Explain the role of subgame perfection in models of (a) collusion and (b) strategic entry
deterrence. What factors make collusion more or less sustainable in a particular industry?
Subgame perfect equilibria (SPE) form a subset of Nash equilibria, particularly used within
extensive form games; subgames in these contexts can be described as a “game within a game”
or a subset of the game. A strategy profile is said to be subgame-perfect if every subgame of the
overarching game represents a Nash equilibrium – this means that at any point, regardless of
what happened prior to this point, the players’ strategic actions should represent a Nash
equilibrium of the continuation game. Particularly, this strategy profile can be determined by
using backward induction in a sequential-moves game, holding the assumption that information
is perfect (i.e. all earlier moves are observed by the players) and complete (i.e. other players’
payoffs are known). These “game plans” should be sequentially rational, meaning that SPEs
exclude Nash equilibria which are upheld by non-credible threats. SPEs can be used to
understand organisational behaviour, particularly in models such as collusion and strategic entry
deterrence, using the example of duopolies to demonstrate a simplified representation of how the
profits (or other forms of outcomes) of firms can be interdependent on the actions of other
players within the market.
Collusion
To begin with, competitors within a market may agree to engage in collusion if doing so would
enable them to generate greater profits. Within a competitive market with only two firms (i.e. the
duopoly as aforementioned), both players would be incentivised to produce at the competitive
price and quantity, namely at p0 and q0 where demand is equal to marginal cost (as the marginal
cost function represents the supply function for a
competitive firm). This is because a firm’s
deviation from this competitive equilibrium in an
attempt to increase profits (namely in favour of the
profit-maximising price and quantity p1 and q1
where MC = MR) would instead generate losses if
its competitor chooses to remain at the competitive
equilibrium with a lower price, as consumers
would simply opt to buy goods from the competing
firm instead, assuming that products are
homogeneous.
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Hertford College
However, this outcome could be avoided if both firms decide to collude, such that both of them
would generate supernormal profits. It should be taken into consideration that for there to be an
incentive to collude, post-collusion profits should be greater than what each firm’s individual
profits would be without the agreement. Firms can do this by forming an agreement to increase
prices to p1 or restrict output to q1. Doing this would enable them to both generate more profits,
at the expense of reduced consumer surplus.
Nevertheless, it could be taken into consideration that even with collusive agreements, there is
still the risk that either firm would cheat and continue to produce at the competitive price level,
such that most consumers would buy products from the cheating firm setting this lower price
level. Such incentives would arise if there is a net increase in profits from the number of added
demands from buyers who switch over from the competing firm, such that the firm who cheats
on the collusive agreement would view a greater increase in profits but would retain a higher
market share than its competitor. This brings rise to a game, such that the concept of SPE can be
used to analyse the firms’ strategy profiles.
Assume that there are two firms in the duopoly with the following payoff profiles:
Firm B
Firm A
Collude
Compete
Collude
5, 5
2, 7
Compete
7, 2
3, 3
As described in the aforementioned scenario, colluding would enable both firms to increase
profits from 3 to 5. However, each firm would also see a greater increase in profits from 3 to 7
(relative to profits under competition) if they cheat out of the agreement and start a price war, in
order to claim market share and potentially force their competitor to exit the market. This payoff
matrix shows that (3, 3) is the Nash equilibrium, as this is the outcome where each player would
have no incentive to deviate from their initial strategy given that their opponent’s action remains
unchanged.
Assume that both firms initially decide to collude. In subsequent periods, either firm can decide
whether to continue colluding or to start a price war. This is illustrated within the chart below: if
Firm A is the first to start the price war, it would gain a greater market share, and Firm B would
be able to choose whether it would also compete to achieve payoffs of (3, 3) or continue setting
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Hertford College
prices or quantities at the collusive level to achieve payoffs of (7, 2). Since 3 > 2, Firm B’s best
response here would be to compete as well. However, if Firm A decides to continue colluding,
Firm B would have to choose to either continue colluding as well – achieving payoffs of (5, 5),
or start a price war – achieving payoffs of (2, 7). If this was a one-shot game, Firm B’s best
response would be to collude. However, within this example, it is also important to note that both
collusion and competition would occur over multiple time periods – resulting in an
extensive-form game with multiple subgames.
The diagram above shows that (3, 3) – where both firms compete – is a subgame-perfect Nash
equilibrium, as it is sequentially rational. Thus, taking into consideration the fact that this is an
extensive-form game, assessing these multiple subgames is necessary to determine how collusion
can be sustained.
Sustainability of Collusion
Understanding these strategy profiles, as well as which outcomes would be sequentially rational,
would also provide firms with insights into how collusion can be sustained by “punishing” any
deviations from the initial agreement. Rees (1993) highlights three such punishments: Nash
reversion, mini-max punishments and simple penal codes.
The example highlighted above has two equilibria – both at competitive and profit-maximising
price & quantity levels. However, some other payoff matrices – particularly those where the
cheating firm’s payoffs would exceed those that it would receive if it continued to collude
alongside its counterpart – may result in only one Nash equilibrium. Nevertheless, in both cases,
the outcome where both firms compete would always be a Nash equilibrium. As it generates
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lower payoffs than the outcome where both firms collude, reverting back to this Nash
equilibrium could serve as a punishment for firms who deviate from collusive agreements. For
instance, if Firm B chooses to compete even though Firm A has chosen to collude previously,
Firm A could “punish” Firm B by choosing to compete as well in the subsequent time periods,
generating lower payoffs for B that could cancel out the gains it received from reneging after
some periods. This is known as a trigger strategy – any deviation would be instantly “punished”
as both firms would subsequently deviate from their agreement to return to the Nash equilibrium
where both firms compete. Additionally, this punishment would hold as it is subgame-perfect –
playing the Nash equilibrium in each subgame would make this strategy profile a Nash
equilibrium of the overarching stage game.
Another form of punishment that could sustain collusion would be imposing a mini-max
punishment. For each individual firm, there would always be a best response action that they
would take given the actions of all other firms. The mini-max payoff would be one where the
actions of all other firms would ensure that an individual firm’s best response payoff would be as
small as possible. This would be the firm’s “security level”, such that they would not accept a
punishment where their payoffs would be lower than this level. Therefore, a mini-max
punishment would be one where the cheating firm would be forced to take the value of their
“security level” – either permanently or until the firm experiences a net loss from cheating. It
should be taken into consideration that the mini-max punishment’s capabilities of sustaining
collusion are dependent on the value of interest rates. Let security level payoffs be denoted by
π0, collusion payoffs to be denoted by π𝐶, and payoffs from cheating to be denoted by π𝐷, where
π0 < π𝐶 < π𝐷.
In order to calculate the net effect of reneging, the firm planning on deviating from the collusive
agreement must compare the one-off gain within the current period, π𝐷 − π𝐶, with the sum of its
future losses in profits from the mini-max punishment, (π𝐶 − π0)/𝑟, with r being the interest
rate for each period. Therefore, the firm would be disincentivised from cheating on the collusive
agreement if the following inequality is satisfied:
π𝐷 − π𝐶 ≤ (π𝐶 − π0)/𝑟, such that
𝑟 ≤ (π𝐶 − π0)/π𝐷 − π𝐶.
However, it should be taken into consideration that this punishment would not be
subgame-perfect. Assume that Firm B cheats on the collusive agreement, and Firm A is now
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Hertford College
faced with the decision of either producing at the profit-maximising level or at the level that
would mini-max Firm B. If Firm B believes that it will receive this mini-max punishment, it will
produce at an output level that would leave Firm A with lower payoffs than it would have if it
were to profit-maximise. Consequently, Firm A’s best response would be to continue to
profit-maximise and not follow through with Firm B’s mini-max punishment. This means that
the mini-max punishment is not sequentially rational, and is held up by non-credible threats.
The third punishment refers to Abreu’s simple penal codes, involving “stick and carrot”
incentives. This would entail a “punishment path” being formed where the firm that deviates
from the collusive agreement would be punished for a certain number of periods until they make
a net loss from cheating (i.e. the “stick”), followed by a reversion to collusion (i.e. the “carrot”).
If the firm deviates again within the first “punishment” phase, the firm would have to restart this
punishment phase. This punishment is subgame-perfect because the threat of punishment is
credible: if Firm B cheats, it would have reason to believe that Firm A would punish it because
not following through with the punishment could lead to Firm A getting punished as well for not
adhering to the collusive agreement.
Apart from these punishments, other factors may also influence the stability of collusion – or,
alternatively, whether the firms would be incentivised to collude in the first place. For instance,
the aforementioned scenarios assume that information is perfect (such that each firm knows what
the other firms have done in previous stages) and complete (such that each firm knows all other
firms’ payoffs as well as its own). In cases where there are information asymmetries or a lack of
transparency within the market, these assumptions may not hold, making it more difficult to
arrive at a collusive agreement as firms would be unsure of what their opponents’ expected
responses would be, and what punishments would be appropriate to disincentivise deviations
from the deal. The previous scenarios also assume homogeneity between the firms; however, this
may not hold in practical cases. Firms may have different cost structures – for instance – such
that lowering prices would benefit a firm with low costs but would harm a firm with higher costs,
making it less feasible for the two firms to arrive at an agreement. Additionally, the number of
firms in the market would play a role as well – in highly concentrated industries with fewer
players, there would be a greater incentive to collude as profits would be distributed amongst
fewer agents. This brings rise to the question of how entry deterrence could be analysed in
assessing competitive dynamics within an industry.
Vanessa Jaya
Hertford College
Strategic Entry Deterrence
The problem of entry deterrence can also be modelled as a game with several subgames, as
outlined by Dixit (1982). Assume that there is an industry with an incumbent monopolist. A
potential entrant would like to choose whether to enter this market or not, taking into
consideration the fact that if it chooses to enter, the incumbent could either share the market or
fight by initiating a price war. The game would have the following payoffs:
Incumbent
Entrant
No Price War
Price War
Entry
pd, pd
pw, pw
No Entry
0, pm
0, p0
The concept of subgame-perfect equilibrium can be used to understand how the incumbent
should formulate its strategy profile to deter entry and retain its market share. To begin with, the
strategy [No Entry, Price War] is not sequentially rational – the incumbent would have no reason
to start a price war if the new firm didn’t enter the market. Therefore, even if (0, p0) is a Nash
equilibrium, it would not be subgame-perfect and adjusting p0 would not be a feasible strategy as
it would be upheld by non-credible threats.
In the aforementioned example, it would be assumed that pm > pd > 0 > pw. Within this example,
it is evident that the incumbent’s profits would be maximised if the entrant stays out, but the
former’s best response if the entrant decides to
enter would still be to share since pd > pw.
Therefore, even though pw < 0, the outcome
[Entry, Price War] would not pose a sustainable
threat to the entrant. Assuming all agents are
rational, using backward induction, the entrant
would know that the incumbent would be better
off sharing the market instead of starting a price
war, incentivising it to enter – this means that
the subgame-perfect Nash equilibrium would be
[Entry, No Price War].
Thus, in order for [Entry, Price War] to be a credible threat, it should be the case that the
incumbent’s payoffs from fighting would exceed those of sharing the market. This can be done in
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Hertford College
the case that the incumbent would make an initial investment of c to prepare for a price war. If a
price war does happen, the incumbent would not have to account for any further costs and would
still have the same payoffs as the initial case (i.e. pw); however, if the price war does not occur,
its payoffs would be reduced by c. Therefore, the threat of fighting would be made credible if pw
> pd - c. The chart below illustrates this new game:
Under this scenario, the entrant would decide not to enter, resulting in the final outcome being
[No Entry, No Price War] with payoffs of (0, pm - c). However, it should also be taken into
consideration that for the incumbent to make this investment, its expected payoffs should exceed
those that it would expect in the initial case where no investment was made and the two firms
end up sharing the market. In other words, the condition pm - c > pd should be fulfilled. Hence,
the concept of subgame perfection enables the incumbent to decide on an appropriate level of
investment to make – the incumbent’s investment c should fulfil the following condition:
pm - pw < c < pm - pd.
Taking into consideration both examples of collusion and strategic entry deterrence, it is hence
evident that the concept of subgame perfection can be utilised to analyse organisational
behaviour and shape the strategic profiles of firms engaging in competitive markets.
Nevertheless, it should also be noted that the scenarios previously outlined rely on assumptions
that may not hold true in practice, particularly the assumption that information is perfect and
complete. Bearing this in mind, firms (and all other agents engaging in similar situations) should
avoid solely relying on the outcomes of subgame-perfect equilibria when establishing
competitive strategies.
Vanessa
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