Two Views on the Second-Mover-Advantage in Game Theory
This essay originated from a paper that analysed the use of Continental Europe's industrialisation as an example of a Second-Mover-Advantage. To make the argument intelligible to the non-specialist, a section was inserted, which aims at explaining the core of basic concepts and the jargon employed in the study of game theory. Readers that are familiar with the subject in question may prefer to skip the first part.
The second section of the essay will attempt to set out, what has up to the present been known as the Second-Mover-Advantage. This is best achieved by analysing the usually cited example where England suffered a First-Mover-Disadvantage during the industrial revolution. It will be shown that in this often praised model there is no advantage for either the first or the second mover.
To clarify this result, the third part will attempt a general definition of a Second-Mover-Advantage. The paper will then focus on one of the rather rare adequate examples in the real world of economics, which fits the pattern of a Second-Mover-Advantage. A less complicated situation close to daily students' life will also be looked at to give a more intuitive understanding.
The final part will seek to clarify the confusion that might have arisen, by giving strict definitions and methods by which games with a First- or Second-Mover-Advantage can be identified.
It should be noted, that the whole of this paper is based on games with two players and two strategies. If not otherwise specified, the analysis is limited to pure strategies.
The author takes this opportunity to thank Dr Richard Green for his valuable comments and much appreciated advice.
Some Basic Notions of Game Theory
In defining a game one can start by looking at parlour games such as chess or Monopoly. It is observed that the fate of one player depends not only on what he does, but is also affected by the actions of the other players. The word ''game'' describes such a situation, where at least two individuals are interdependent on one another. This definition, however, also includes situations, that in ordinary English would not be classified as ''games''; competition for status, arms races, international treaty negotiations or the operation of market economies may be cited as examples. Game theory can then be seen as the study of multi-person decision problems.Formally, a game is comprised of (i) a list of participants, or players, (ii) for each player, a list of strategies and (iii) for each combination of strategies a list of payoffs that the players receive (the first figure generally represents the first player's payoff).
Fig. 1(a) depicts such a game in normal form, i.e. no attention is given to the timing of actions that players may take. The participants choose their strategies (plans A or B) without knowing of the other's choice. In this particular game player 2's second strategy is strictly dominated by his/her first, i.e. plan B yields a higher payoff than plan A, no matter what strategy player 1 may choose. Due to the assumption in game theory that all agents behave rationally and reason optimally (they are said to be homines economici -- see Note 1), it can be asserted that payoffs (0; 1) and (4; 2) are impossible. Player 1 knows this (he/she thinks rationally too, and can pretend being in the opponent's situation), and hence chooses plan A, the payoff of which (7) is higher than that of plan B (3). In this equilibrium situation no player has an incentive to deviate from his/her plan; every player's strategy is the optimal response (giving the highest payoff) to the other participant's actions. Such a strategically stable and self-enforcing equilibrium is called a Nash equilibrium. Nash equilibria are not necessarily intuitively obvious, nor do they always display social desirability.
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Fig. 1(a) | ||
If a game contains more than one Nash equilibrium, than the extensive form representation shown in Fig. 1(b) might be more useful than the normal form. In the extensive form the players do not have to make their decisions simultaneously, but rather the second mover may choose his/her strategy with knowledge of the other player's decision. In the game discussed above the sequence of moves is of no importance due to the unique Nash equilibrium.
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Fig. 1(b) |
However, when the payoff (4; 2) is modified to (4; 10), two Nash equilibria are observed: (7; 9) and (4; 10). After the modification the order of moves will determine the actual outcome of the game. Player 2 prefers the latter equilibrium while player 1 favours the former. Whoever moves first, will be able to choose his/her favourite equilibrium. (There may, of course, be situations when both players prefer the same Nash equilibrium, for instance if (4; 7) is substituted for (4; 2).)
In analogy to the First-Mover-Advantage the concept of the Second-Mover-Advantage can be identified. In this author's view this idea has often been developed in a misleading if not wrong manner. The criticisms of the traditional approach and an alternative model put forward by the author are discussed in the following sections.
A Traditional View on the Second-Mover-Advantage
Not infrequently it is said that in a particular game the second mover enjoys an advantage. It is all the more difficult to find a thorough examination of such a situation. For most authors, it suffices to give an example, which in this author's experience has commonly been the industrial revolution. The following (see Note 2) constitutes a brief version of the model, in which England has a First-Mover-Disadvantage. In a game with two players (England and Continental Europe) both parties have the choice between industrialising or not doing so. The payoffs are as set out in Fig. 2(a). It is supposed that England has to move first. Her payoff can easily be determined as 20, whereas Continental Europe obtains 45. It is then claimed that Continental Europe enjoys a Second-Mover-Advantage. Any further explanation is considered superfluous. The result is generally taken for granted without thorough questioning.
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Fig. 2(a) | ||
However, upon closer inspection the game turns out to be no different from games such as the one represented in Fig. 1. Yet there, nobody has ever suspected a Second-Mover-Advantage. Returning to games with an advantage, from situations with a First-Mover-Advantage one can deduce, that if the sequence of moves is reversed, the advantage moves from one player to the other. As a necessary consequence the outcome must change and the equilibrium payoffs cannot stay the same. Applying these conditions to the above game one obtains some surprising at first, but irrefutable results. No matter whether England moves first or second, the outcome stays at 20 for England and 45 for Continental Europe. Hence it cannot be correct to maintain that there is a Second-Mover-Advantage of any sort. The game in Fig. 2(a) constitutes a game with one single Nash equilibrium, for which case the equilibrium outcome cannot change -- the same is the case with the ordinary game in Fig. 1.
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Fig. 2(b) |
Of course, common sense would suggest the invalidity of the above statement; after all, is it not self-evident that, had Continental Europe industrialised first, England might have learnt from the mistakes across the water, and hence achieved results higher than the 20 in the above case? However, what is often forgotten in this debate is actual game theory, part of which postulates that a game is defined by its payoffs. Clearly, the payoffs in our first example (cf. Fig. 2(a) ) only fit the sequence of England moving first. Should this be changed, then the payoffs change -- which probably has led some game theorists to believe in a Second-Mover-Advantage -- but with different payoffs the game changes. What has been done in the past, is comparing two different games, while believing that a single game with a Second-Mover-Advantage was being observed. Already the definition of a game makes this line of argument fallacious. Considering each of the two games in turn one can find a single Nash equilibrium respectively, thus not leaving any reason to put forward a Second-Mover-Advantage.
A Real Second-Mover-Advantage
The above analysis is of course not to say, that there is no such thing as a Second-Mover-Advantage. What it does show is the impossibility of that phenomenon in the case of at least one Nash equilibrium. A Second-Mover-Advantage can only, or rather must occur when there is no Nash equilibrium (see Note 3). This theorem is not difficult to show in general terms, but an example will suffice (cf. Fig. 3). (While the next paragraph illustrates a case in the world of economics, the key insights into the real Second-Mover-Advantage are not lost by moving immediately to the last paragraph in this section).In this game there are two firms, call them Piccolo and Gigantic. They both produce similar products; however, Piccolo has only a small market share, while Gigantic controls a large proportion of the market. Both firms want to bring a new product onto the market, for which they must introduce one of two systems. But Gigantic has rather inferior technology compared to Piccolo. If both firms operate with the same system, then the whole market is in that very system, and can choose between the two technologies. It will presumably favour Piccolo. On the other hand, if the two firms operate different systems, Gigantic's customers are stuck with Gigantic's system and hence have no access to Piccolo's advanced technology, which hence can only be sold to Piccolo's market share. According to this rationale the payoffs in Fig. 3(a) are set.
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Fig. 3(a) | ||
It will be discovered that there is no Nash equilibrium. One will further observe that in spite of the absence of such an equilibrium it is possible to solve the game as long as the sequence of moves is known. If Piccolo moves first, the outcome will be (3; 6). Gigantic would respond to Piccolo having favoured system A by choosing system B (since 6 is a higher payoff than 4). If, on the other hand, Piccolo had chosen system B, Gigantic would react with system A (7 is higher than 5). Piccolo, knowing this, will select system A (3 is higher than 2); the final outcome is hence 3 for Piccolo and 6 for Gigantic (This way of solving games is called ''backward induction''). Similarly, it can be determined that the oucome changes to 9 for Piccolo and 5 for Gigantic, when the order of moves is reversed. Inspection of the outcomes shows clearly, that both Piccolo and Gigantic prefer the equilibrium outcome in the case when they move second respectively; the second mover enjoys the Second-Mover-Advantage.
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Fig. 3(b) |
In a rather simpler example closer to daily life, suppose two students, Tony and Mary, attend the same school. However, while Tony dislikes Mary, Mary is madly in love with Tony. When it comes to lunch time, they both go to the same dining hall, where there are two big tables. As Tony does not like Mary, he wants her to sit at a different table; Mary on the other hand wants to spend as much time as possible near her beloved Tony. Clearly if Tony sits down first, the two of them will end up sitting at the same table, which is Mary's preferred outcome. If Tony moves second, he will not sit at Mary's table, so he gets his preferred outcome. The second mover will be able to pick his/her/its preferred outcome, and thus enjoys an advantage.
Appendix: Mixed strategies
The rows and columns in a bimatrix game are called pure strategies to discern them from mixed strategies. A mixed strategy is used when a random device is employed in determining what to do in a game.The following constitutes an example for a game, that possesses a Nash equilibrium only in mixed strategies. Recall the two students Tony and Mary. Suppose that for meals tickets must be purchased in advance; there are tickets for both tables available. As long as this game is played without the tickets, the outcome can be established with reference to the sequence of moves. If, however, both Tony and Mary must buy their tickets in advance, they will not know of the other's choice. In mixed strategies the two players may well find it optimal to toss a coin in deciding on a table.
This result is based on the assumption of optimal reasoning in game theory. Any line of argument that a rational player may follow can be understood by the rational opponent. Thus by duplicating Tony's thoughts Mary could easily predict the table he will be sitting at. However, by way of a reductio ad absurdum Tony will realise that Mary comprehends his reasoning. Hence the only solution for a rational player in preventing his/her choice of pure strategy from being predicted by the opponent is by resorting to a randomising device. In other words, each participant in the game aims at preventing the other from predicting his strategy, and nothing can be harder to predict than a decision made at random.
Notes:
- On a slightly pedantic note, the author apologises for the etymologically incorrect spelling; it appears to have become adopted this way. Being derived from the Greek oikos the legitimate spelling is oeconomicus.
- This example is taken from lecture notes as handed out by Professor Partha Dasgupta during Lent Term 1998 in the University of Cambridge.
- In mixed strategies there always exists at least one Nash equilibrium; hence, in order to determine the existence of a Second-Mover-Advantage, only pure strategies must be considered. In general therefore, mixed strategies are not relevant to the analysis in the main body of this paper. However, the reader may be interested in the outcome of a simultaneous-move game with no Nash equilibria in pure strategies, if mixed strategies are included in the analysis. The appendix attempts such an investigation.