A short introduction to game theory

In game theory, we are usually looking for one or more equilibria (ideally only one), which we regard as representing the likely outcome of a particular situation. The principal criteria which an equilibrium is expected to satisfy are the Nash equilibrium condition and 'subgame perfection'.

Dominant strategies

We can write the strategy of player i as s_i. By strategy we mean a particular move or policy, e.g. “produce low output” (collude), or “cooperate if and only if the other player did so the previous round” (trigger).

We can write the payoff to player 1
if player 1 plays s₁ and player 2 plays s₂
as u₁(s₁,s₂).

If u₁(s₁^A,s₂) < u₁(s₁^B, s₂) for all possible s₂ and some s₁^B
(i.e. whatever player 2 does, it is possible for player 1 to do better than s₁^A)
then s₁^A is a dominated strategy.

If a strategy remains after iterative removal of all dominated strategies, it is a rationalisable strategy.

If there is only one rationalisable strategy, it is the dominant strategy (e.g. “defect” in the Prisoner's Dilemma game).

Nash Equilibrium

For n players, (s₁*,s₂*,…,s_n*) is a Nash equilibrium (NE) if and only if:
u_i(s₁*,s₂*,…,s_i*,…,s_n*) ³ u_i(s₁*,s₂*,…,s_i,…,s_n*) for all s_i and all i.

All NEs consist of rationalisable strategies, but not all combinations of rationalisable strategies are NEs.

All combinations of dominant strategies are NEs, but not all NEs consist of dominant strategies.

Mixed strategy equilibrium

A mixed strategy is a probability distribution over strategies. I.e. a given player is playing each of his possible strategies with some probability. For example, firm A produces high output with 40% probability and low output with 60%, and firm B similarly mixes these strategies, but in the ratio 70:30. The game may be one-shot, so the point isn’t necessarily that the players alternate between strategies.

In this case the NE consists of optimal probability choices by each player given the probability choices of other players.

Once you allow for mixed strategies then every (finite) game has at least one NE.

Non-cooperative game

In a cooperative game, the rules permit binding agreements prior to play. (Hence collusion would be possible in a one-shot cooperative game.) In practice, we are usually concerned with games in which this is not possible, i.e. a non-coooperative game.

Games of complete information

• Players’ payoffs as functions of other players' moves are common knowledge.
• Each player knows that other players are 'rational' (i.e. payoff-maximising), and knows that they know that he is rational.

Static and dynamic games

In a static (or ‘one-shot’) game, players move simultaneously and only once. In a dynamic game, players either move alternately, or more than once, or both. Bertrand and Cournot models of oligopoly competition are both static games, while the Stackelberg model (firm B moves after firm A) is a dynamic game.

An equilibrium for a dynamic game must satisfy subgame perfection.

Normal and extensive forms

A game expressed in ‘normal form’ is in the form of a payoff matrix, as shown below for the Prisoner's Dilemma game.

A game expressed in ‘extensive form’ shows the ‘tree’ of the possible move paths depending on what each player does at each stage. A dynamic game can only be shown in extensive form.

Repeated games

The same agents repeatedly playing a given one-shot game (“form game”) in sequence is called a supergame. A supergame can consist of either a finite, or an infinite, repetition of a form game. Or we could have a game with a certain probability p of being repeated, i.e. a probability 1 – p of breakdown.

A strategy in this context can be contingent on what other players have done in previous moves.

Subgame perfection

For a dynamic game, some Nash equilibria are not acceptable as solutions because one or more players will want to, and be able to, avoid those outcomes. The subgame perfection criterion demands that, at each stage of the game, the strategy followed is still optimal from that point on.

Non-credible threat

A 'non-credible threat' is a strategy that one player is trying to use to manipulate the behaviour of another (usually via the second move in a sequential game), which forms part of a Nash equilibrium but one that is not subgame perfect.

The strategy is one which is being claimed in some way by the threatening player (e.g. by means of a signal that he is capable of using it) but which is not credible: although the threatened player’s optimal response to the strategy is to do what the threatening player wants, the former knows that by moving first in a different way, the latter will adopt another strategy to generate a different Nash equilibrium.

In a sense, there is no ‘credible threat’ — the term ‘threat’ implies that player 1 will do something specifically designed to harm player 2 if player 2 doesn’t comply, but such a threat would never be carried out in a finite game with perfect information because such a move would not be optimal for player 2. Player 2 will always ‘accommodate’ when it comes to it.

[Next week: applying game theory to Northern Rock & the Bank of England]

Depreciation and price regulation

Price regulation, for example in the case of a monopoly supplier, often involves determining an acceptable rate of profit. Profit is normally calculated after ‘depreciation’ i.e. taking account of the wearing out of capital assets. It is sometimes suggested that, since depreciation is not a real cost, actual capital expenditure should be used instead to determine the real profit level under different output prices, and hence the acceptable output price. In this article I argue against this approach.

1. The purpose of depreciation

According to Financial Reporting Standard 15 issued by the UK’s Accounting Standards Board, the objective of depreciation is “to reflect in operating profit the cost of the use of the tangible fixed assets (i.e. the amount of economic benefits consumed by the entity)”. The Standard adds that depreciation should be allocated to accounting periods in a way that reflects “as fairly as possible the pattern in which the asset’s economic benefits are consumed by the entity.”

The purpose of depreciation is therefore not to accumulate reserves to finance the future replacement of the assets which are being depreciated. The purchase of an asset is an expense for a company. The choice of accounting treatment is between a full write-off against profits at the time of purchase, and a gradual write-off over the useful life of the asset. In neither case is a corresponding fund set up for eventual replacement.

This view of depreciation, that it represents past rather than future expenditure, is reflected in taxation law. Some form of gradual write-off of past capital expenditure is usually allowed as a deduction in calculating taxable profit. Transfers of profit to fixed asset reserves, on the other hand, are not permitted as deductions. In the UK, this is true both for corporation tax and for petroleum revenue tax. Most other countries’ tax regimes share the same view of depreciation.

2. Return on capital expenditure

The return on a company’s investment in capital expenditure is two-fold. First, the company expects to recoup the capital expenditure over the life of the relevant assets. If it does no more than that, it is simply breaking even. Second, it expects to earn revenue over and above this break-even level during the life of the assets. This additional revenue represents its profit, and is the return on the capital employed.

The first, merely neutral, element in the return on investment is taken into account by deducting depreciation in the calculation of real profit. Hence if depreciation is excluded as a cost in calculating profit, a misleading figure for return on capital is obtained.

3. Financing of asset replacement

There are two principal sources of finance for companies. The more important of the two is internal finance, i.e. a company using its own reserves, represented by cash or short-term investments. Alternatively, a company may obtain external finance, either equity (typically by means of rights issues) or borrowing. Most finance, particularly for fixed asset replacement, is internal.

Typically companies replace fixed assets gradually each year as they wear out, and the asset replacement profile is relatively smooth over time. Since real cashflow exceeds accounting profit by the amount of the depreciation charge, this annual undistributable cashflow excess can be used to finance annual asset replacement. The match between depreciation and fixed asset replacement expenditure is likely to be reasonably close, especially if the current cost accounting form of depreciation is used.

Where the asset replacement profile is not smooth, e.g. where a large asset base is expected to wear out during a relatively narrow time-window (as may happen e.g. with oil or gas pipelines), there are two choices for what to do with the undistributable cashflow excess represented by depreciation.

First, it can be accumulated over a period of years in the form of cash or liquid investments for the purpose of eventual investment in fixed assets. Secondly, the accumulating funds can be invested in long-term projects or business operations in such a way that the funds are potentially ‘tied up’. In that case, external finance may have to be raised when the time comes for the programme of asset replacement. However, subject to capital market imperfections, this should be as efficient a way of financing the programme as the internal accumulation of funds.

A criticism of the first of these two possible approaches is that the purpose of the company from the point of view of its shareholders is to invest available shareholders’ funds in business activities which will earn a better return on those funds than shareholders could do for themselves.

Accumulating large cash or short-term investment reserves is not usually considered appropriate for a company. Companies with large cash reserves are often under pressure from shareholders and analysts to eliminate the reserves in order not to dilute return on capital employed, by using the funds for expansion.

4. Fixed asset reserves

The use of fixed asset reserves to accumulate funds specifically for the purpose of future fixed asset expenditure is not a common business practice in the UK, nor indeed in the rest of Europe or in the US. On the other hand, a company with good financial management will inevitably plan for future cash requirements by appropriate build-up of cash levels or by arranging borrowing facilities in advance.

The closest analogy in UK commercial practice is the use of so-called ‘captive insurance companies’, which are effectively ring-fenced funds designed to provide financial cover for the kinds of eventuality normally insured against, but without the owner of the captive insurance company losing ultimate control over the insurance monies.

5. The effect of price regulation

Where price regulation is based on a target rate of return on capital employed, it is sometimes questioned whether depreciation should be taken into account as an expense in calculating permitted revenue levels. For example, it was argued in relation to TransCo* that, to the extent that depreciation in a period is not matched by expenditure on fixed assets in the same period, allowing depreciation as a cost results in an excessive permitted revenue level.

To the extent that allowed revenue under the depreciation-based approach to setting revenue would exceed allowed revenue under a pay-as-you-go approach [i.e. including current capital expenditure among costs which revenue has to cover], revenue could be considered to be provided to TransCo in advance of its cash requirements. **

However, this argument fails to take into account the point about depreciation made above, namely that it represents a return of initial capital which must be covered by revenue, in addition to any ‘return’ on capital in the sense of profits, for the business to meet its objectives.

Saying that any excess of depreciation over fixed asset expenditure represents a kind of distortion can be used to argue that a justification must be found if the distortion is to be permitted. In the case of TransCo, one justification which has been proposed is that a build-up of such excesses is required to fund an eventual reversal of the situation, i.e. that in due course capital needs for fixed asset replacement will exceed depreciation charges. The authors cited above argue against this by claming that, in view of the difficulties in predicting future required fixed asset expenditure,

the uncertainty associated with the level of future capital spending may well mean that the case for any revenue advancement is weak.

This is an unsatisfactory argument since what is here called ‘revenue advancement’ is not an active process requiring judgments about the appropriate levels of advancement, but rather a case of passively allowing the existing excesses of depreciation charges over fixed asset expenditure to build up in anticipation of a future investment programme. Uncertainty over the precise amount of expenditure involved in this programme is not sufficient to undermine the validity of such ‘revenue advancement’.

One argument in favour of allowing depreciation rather than expected capital expenditure as a cost in calculating permitted revenue is that the resulting price cap is likely to be much less stable under the latter method. Annual depreciation typically has a much smoother profile over time than capital expenditure.

* now part of National Grid plc

** Arthur Andersen, TransCo 1997 price review, 3.6.

Charts of the month

These are two charts from Marc Faber's recent lecture 'Gloom, Doom, or Boom?'. The first (source: Ned Davis Research) shows that the last time US debt grew to such stratospheric levels as now was in the 1930s.

The second chart (source: Stifel Nicolaus) suggests that we are due for an upturn in consumer inflation.

Will the internet increase monopolisation?

The internet has been enthusiastically proclaimed as a harbinger of greater competition. The work of Thomas Malone, among others, has predicted an 'electronic market' effect. The results of this are alleged to generate a highly competitive, rivalrous business world.

Prima facie, a reduction in transaction costs might be expected to facilitate marketisation of business-to-business (B2B) commerce and to reduce lock-in. The internet allows information to be both posted and accessed cheaply and easily, making it possible for companies to 'shop around'. In theory, this will make real attributes such as price and quality more important than advertising or reputation, giving small firms a better chance and increasing competition. Companies will find it easier to outsource, but will be less likely to form close integrated relationships with their suppliers, preferring to form casual, temporary relationships via the Web. This summarises the orthodox view of the effects of the internet on commerce.

It is often forgotten, however, that competition in Marshall's sense — i.e. rivalry between homogeneous firms — is an incidental by-product of capitalism. It depends crucially on it being efficient to have a particular activity carried on by a large number of firms rather than by a few. In the long run, this key condition may fail to hold. To consider the likely effects of the internet on market structure, we need to consider how improvements in information exchange will affect efficient firm scale and scope. This topic has not been explored in detail in either the economics or management literatures.

The issue of firm size is one of the key questions of industrial organisation. It is also one which has been linked since the early days of industrial economics with the issue of competition — Knight (1921), for example, argued that the force arising from the powerful desire to benefit from a monopoly position 'must be offset by an equally powerful one making for decreased efficiency.' The issue of efficient size more recently came back into attention, with some (e.g. Arthur 1996) arguing that decreasing returns to scale have ceased to be a relevant feature for many industries, and that in some we are seeing increasing returns.

The question of efficient scale determines concentration, and hence competition and static efficiency. The question of efficient scope affects how focused a firm is — i.e. how many ancillary activities it internalises — and therefore determines the boundaries of firms. The restructuring of business over the last fifteen years — outsourcing, globalisation, mergers — suggests that a change in conditions may have led to a change in what is efficient. A number of researchers (e.g. Malone, Yates and Benjamin 1987, Hammer 1999) have already observed and documented the link between outsourcing and B2B.

Each of the issues of scale, scope and overall firm size need to be considered. With regard to scale, when different companies operate in the same area of business, some duplication of costs is inevitable. Prima facie, therefore, there are potential efficiency gains (economies of scale) from expanding market share, or by simply integrating with rivals. In theory, one should always be able do at least as well by having two similar businesses under common ownership. Since firms tend not to expand without limit, however, there are clearly countervailing forces at work. These have usually been supposed to operate at the managerial level.

The issue of scope has received considerable attention through the work of Coase, Williamson and the property rights theorists (e.g. Grossman and Hart 1986). The key reason for internalising processes secondary to a business's core activity is generally agreed to be that of 'transaction costs' — the incidental costs of contracting with independent parties. Most recently, these costs have come to be identified principally with inefficiencies that arise ex post because contracts are incomplete and therefore not capable of providing ex ante protection against opportunism. As the incomplete contracts literature has emphasised, even a watertight contract will not always be honoured, and legal measures after the fact are expensive and can be ineffective. These theories would explain the current increase in outsourcing as due to improvement in contracts and a lessening of opportunistic behaviour. However, as Brynjolffsson (1993) has observed, empirical studies (e.g. Kanter 1989, Piore 1989) do not support this interpretation. I propose that other types of transaction cost are more important in answering this question, most notably losses arising from imperfect information exchange. Subject to some exceptions (e.g Radner 1996, Casson and Wadeson 1998), these have received relatively little attention to date as a source of transaction cost.

Finally, on the question of the overall size of a firm, it has generally been agreed that the management function typically yields decreasing returns to scale above a certain size, and that this is one of the main reasons why diseconomies of scale set in as firms expand market share. This may be due either to the fact that expanding firms require increasing spans of management, or to the information costs of communicating between different divisions in different locations. One way of interpreting these effects is by reference to bounded rationality — Williamson (1985), for example, summarises this view by arguing that effective expansion ceases when "bounds on cognitive competence" are reached.

Putting these three factors together yields a theory of optimum firm size and scope, which is a function of the trade-off between (a) production economies of scale which favour 'wide' firms, (b) savings in transaction costs from internalising ancillary functions which favour 'deep' firms, and (c) managerial diseconomies of size which favour smaller firms. Effects (a) and (b) each make firms want to expand — horizontally and vertically, respectively — while effect (c) restrains them from doing so. Thus, the need to keep managerial costs down can be achieved either by (i) reducing scale, which means economies of scale are lost and average production costs rise, or (ii) vertical disintegration, which increases transaction costs.

The motive for outsourcing is often said to be that it enables a business to concentrate on its key activity, or 'core competence'. Provided that outsourcing can be achieved with a relatively low resulting increase in transaction cost, it enables businesses to exploit economies of scale that would previously have been offset by managerial diseconomies. I argue that electronic information exchange (EIE) has had, and will continue to have, the effect of facilitating outsourcing in order to exploit economies of scale.

One consequence of this observation is that if transaction costs generally fall, there will be a tendency to switch away from (i) in favour of (ii). In other words, horizontal integration will take place at the expense of vertical integration: outsourcing facilitates mergers.

Little work has been done on explaining the recent trend for increased concentration as a result of horizontal mergers, although Chandler's work suggested that decentralisation is more feasible for the firm with greater lateral scope. Chandler (e.g. 1962) argued that functional scale economies are not great enough to make up for the diseconomies caused by the co-ordination problems inherent in functional structure, and that large firms can combine the most advantageous features of both if they undergo a structural shift into divisional management. General Motors and DuPont are pioneering examples of this. It would seem to be easier for horizontally integrated firms to undergo this shift without incurring inefficiency caused by communication errors than for vertically integrated firms.

One indirect consequence of the tendency of sectors to become monopolised as a result of increasing exploitation of scale economies is that bilateral trade (i.e. exclusive trade between two specific parties) becomes a more attractive option, simply because there is relatively little competition. This helps to explain why we are seeing increasing talk of buyer-supplier 'partnership', and related concepts increasingly featuring as key aspects of supply chain management. Corporations such as Bose practise hole-in-the-wall manufacturing: manufacturers make their products on site in Bose's factories and participate in the company's management meetings. Bose also pioneered 'JIT2', involving suppliers permanently on site, a practice that has spread widely since it was first introduced in the early nineties. Companies, rather than keeping their options open with a large supplier base, are cutting the numbers of their suppliers. Companies seem happy to be closely tied together, and are using e-business to build closer links with their suppliers.

Malone, Yates and Benjamin (1987, 1989) deny this effect in favour of the theory of increased marketisation. They predict that short-term buyer-supplier contracts, not partnership, is the principal effect of EIE — or at least of the form of it with most potential flexibility, the internet. It is true that in areas where homogeneous, irregular inputs are required, the use of web features like electronic auction is likely to lead to more static rivalry. By reducing search costs and rendering information accessible, the internet may indeed make a firm's search for a new supplier more market-like. However, it is important not to mistake a number of special cases for the whole picture. Overall, the evidence suggests increasing concentration and increased use of partnership. Companies appear to prefer partnership relationships where possible because they "improve quality and performance and [...] reduce costs" (Dussuage and Garrette 1999). Steinfield et al. (1998) find that electronic communications networks are being used more often than not as part of a strategy of tighter buyer-supplier relationships.

If e-business does not on balance increase marketisation, but rather encourages monopolisation and lock-in, we need to consider the effects on competition and static efficiency. I suggest that Malone et al's vision of the principal impact of e-business — internet-engendered markets giving easy access to a wide range of facts on different suppliers, benefiting industry and ultimately the consumer by keeping prices low and quality high — is flawed. Horizontal merger activity is likely to continue to feature at every level of business, further increasing concentration.

However, the conventional model of competition — several suppliers of the same product battling for the same customers — is likely to be increasingly replaced by more dynamic forms of competition in which one firm seeks to displace rivals by more efficiently matching customers' needs. Furthermore, potential competition, of the kind proposed by contestable market theorists (e.g. Baumol 1982), in which the behaviour of incumbent firms is constrained by the threat of entry, is likely to replace static rivalry. This will happen as entry barriers decline because firms increasingly focus on core competences and carry out smaller and smaller ranges of activities. A horizontally integrated, vertically disintegrated industry of the kind we have outlined maintains efficiency through fiercer potential competition. It is far easier for a rival to set up in competition with a flat, specialised market niche player than it is to set up a whole value chain in competition with a vertically integrated firm.

As Baumol, Panzar and Willig (1982) have argued, concentration of output need not be harmful so long as barriers to entry are low: the mere threat of competition can make a firm behave competitively. Microsoft, for example, has a near-monopoly in many areas of PC software but arguably remains innovative because its markets are contestable. Electronic information exchange lowers communication costs and thereby also helps to lower barriers to entry.

This is a summary of a paper I wrote in 2000. I have not had an opportunity to update it for any recent developments. A version of the full paper, including a simple numerical model, empirical data, and bibliography, is available here.

Such a thing as society

Is there such a thing as 'society', over and above a collection of individuals? Curiously, the question is now associated in many people’s minds with Margaret Thatcher, who said:

I think we have gone through a period when too many children and people have been given to understand "I have a problem, it is the Government's job to cope with it!" or "I have a problem, I will go and get a grant to cope with it!", "I am homeless, the Government must house me!", and so they are casting their problems on society and who is society? There is no such thing! There are individual men and women and there are families and no government can do anything except through people and people look to themselves first.

The argument may of course originally have been mentioned to her by one or more of her intellectual supporters. It has also been pointed out that it was made in a fairly informal context, i.e. an interview with Woman’s Own (in 1987).

There are several ways of interpreting the question. Unfortunately, attention tends to be focused on the least interesting one. The trivial answer is, “yes, of course there is”. The word is useful in the same way that the concept of ‘global warming’ is useful — it is a shorthand for describing complex phenomena by means of aggregating and averaging.

Probably one can go further, and say that there are social phenomena which, while ultimately reducible to individual behaviour, only become observable at the aggregate level. They are, in other words, a kind of (weakly) emergent behaviour. For example, to say that "Italy advanced technologically in the sixteenth century" is theoretically reducible to the behaviour of individual Italians, but it would be difficult to do so, and the aggregate phenomenon is not readily predictable simply from information about the individuals.

However, to say that patterns emerge at an aggregate level which are not conveniently reducible to descriptions of individual behaviour is not the same as saying that a different kind of entity emerges at that level. There is clearly a sense in which methodological individualism — the claim that the whole is ultimately nothing but the sum of its parts — is trivially true, and anything more ambitious than weak emergence — e.g. social holism — is 'obviously' false.

The arguments that individuals are influenced by their social environment (situationism), or even that they are largely a function of that environment (constructionism), are really concerned with a different issue from the one raised by Thatcher's assertion. Even if the actions of individuals are always traceable to societal influences, those actions are still the basic components of social behaviour, and the latter must ultimately admit explanation in terms of the former.

However, all the above points are somewhat incidental to the issue typically at stake when the Thatcher quote is discussed. Reactions to it are more often concerned with a different matter altogether: that of whether ontological or moral priority should be given to the individual over the social group. In other words, the argument is really about the conflict between political individualism and communitarianism.