From Ben's Writing

A paper for Philosophy 3411 - Game Theory [Fall 04]

Contents 1 Introduction 2 Definitions 3 Synchronization 3.1 Locks 4 Games as agents [6] 5 Footnotes

Introduction

In a paper by Viminitz entitled Games of Pure Paralysis, we are offered a proof from which, if we accept Viminitz’s analysis of the situation, we must accept that there may be a flaw in our existing theory of games. The game Viminitz uses is derived from the movie The Deer Hunter, and is aptly named: The Deer Hunter Paradox (DHP). What concerns us is a particular scene in which the main characters are in the rather disagreeable position of been forced to play out rounds of Russian roulette as entertainment for their hosts, the Vietcong (VC). To make a long story short, the proof illustrates a possible problem with the current game-theoretic model. If true, it suggests that, given the protagonist’s situation, it is rational for him to rebel against his captors given a gun loaded with a single bullet ^[1].

This paper is not intended as a solution to the DHP; rather what it offers is a method by which we can better understand the mechanics of the game. The initial ideas outlined below are not my own, though the expansion on them is. The two main ideas are as follows: First, there is the insight that the DHP game is actually a composition of more than one game ^[2]. To be more precise, there is the game of Russian roulette (RR) which the VC are forcing upon the protagonists, Mike and Nick, and there is the game of Fake-out, which Mike devised in order to disguise the rebellion, the game of Shootout, he and Nick have planned. The second idea is the suggestion that there might be an overlap in these games^[3]; that is to say, a play in one game may result in a change in the structure and outcome of the other. These ideas are interesting in their own right; however, we will concentrate on how they relate to a parallel model of game play.

Definitions

Unfortunately, the definitions given in the available literature are full of contradictions so we define our meanings here:

1. A resolution ^[4] to a game is an assignment of a strategy to all agents in the game from their domain of strategies.
2. An equilibrium point (EP) is a resolution to a game such that the strategies assigned to each agent maximize that agent’s payoff if the strategies of the other agents are not changed.
3. The resolution space (RS) of a game is the set of all possible resolutions to that game.
4. A shared resolution space (SRS) is an overlap in the resolution space of two or more games.
5. An overlapping game (OG) is a set of games that have shared resolution space.
6. A parallel-composite game (PCG) is an overlapping game in which the subordinate games are played concomitantly.
7. A model for parallel play

To understand the implications of these ideas we will first need to define a new model that supports this new parallel mode of play. As with any model, several problems must be tackled in order to ratify it. First, there is a need for a synchronization mechanism to restrict—or rather, referee—access to the shared solution space. Second, if there is a need to preserve the predictability of the game, we may need a stricter version of the above-mentioned mechanism. We shall return to this second point later but for now, let’s deal with the basic requirement of synchronization.

Synchronization

It is intuitively obvious that we can have overlapping games. Look at the game of basketball for example. Here we have a game composed of two games: the cooperative game between team-mates; and the non-cooperative game between teams. Of course, it is possible to have a game of basketball that is strictly non-cooperative, as in the case of a one-on-one match, but this does not rule out the possibility for overlapping games.

The method for composing games is simple. It can be accomplished by passing the outcome matrix of one game as a parameter to another. It should be noted that this technique is non-commutative; the composition g1(g2()) is not equivalent to g2(g1()). This non-commutative property is the major complication for the parallel model of play. To illustrate the problem of concomitant access of the SRS, let us think of two children reaching for a candy-bar. If child A gets it first, then child B will get nothing, and vice versa. Now imagine these two kids reaching the chocolate bar at once. Keep in mind that these children are young and have not learnt the concept of sharing yet. What happens to this chocolate bar? It is hard to tell, really. They could tug at it hard enough so that the wrapper bursts and they each get some portion. Alternatively, it could be that one child is stronger than the other is, so she gets to keep it all. Etc. The point is that, while the effects of playing g1 before g2 in the classic model are predictable, this is not the case in the parallel model. Well, without a little help, that is.

As we have seen, the problem for these new games lies in the temporal relation of its subordinate games and their concurrent access of the SRS. The possibly unrestricted access of the SRS may have the effect of permuting the SRS in such a way that it gives rise to a new breed of undeterminable games.

In contrast to a classic game an unstable game (UG) is one in which given the strategies of the agents the payoff matrix cannot be computed. This anomaly is due, in part, to the concurrent access of the SRS. With no method in place to restrict access, we run the risk of leaving the SRS in an undefined state. So, while making a play in a classic game affects only the RS of that particular game, a similar play in a PCG may have an affect on one or more resolution spaces. This means that if we were to have multiple rounds of the same game under the same circumstances and stipulating that the agents adopt strategies identical to the previous round in each subsequent round, it would not necessarily be true that each round would have an identical resolution.

Surely, this cannot be the case. If ones’ actions in a game have no direct influence on the outcome, then there is no reason to suspect any agent would be willing to take part in such a game. What we need is a method to encourage agents to play, a way of stabilizing the game.

Locks

One way of achieving this would be to construct a locking mechanism by which to we could restrict access to the SRS—restricted in the sense that only one game can gain access to it at any one point in time. However, here too there are several problems:

To begin with, let us explore the option of not having a locking mechanism. We do this to illustrate the importance of it in our new model. If no mechanism to lock the SRS existed, these types of games will prove to be impossible to analyze. The reason for this is simple; if there is unrestricted access to the SRS, multiple games might alter the SRS at the very same point in time leaving it in an undefined state. This poses a problem for agents, as it would be no fun to play a game in which a straight forwardly maximizing move on the part of an agent resulted in a non-maximal position ^[5].

Now, let us make some locks.

First, we shall consider a fully functioning and game specific lock. Such a lock would be endowed with all the information required in order to arbitrate the PCG it was tailored for. This type of lock would save us from this nondeterministic property of PCG games; unfortunately, it sounds rather too good to be true. We will not, however, discount it just yet, for as we shall see later, it is possible that such a lock might exist.

Secondly, we consider a ‘faulty’ version of the above. This lock would be completely generic and contain no specialized information. Its sole purpose is to decide, albeit arbitrarily, which game gets access to the SRS and when. This seems a far more palatable answer, as we need not invent anything new: the effect can be observed in the toss of the coin or the spin of a roulette wheel, “Around and around it goes. Where it stops, no one knows.” Unfortunately, this does not solve the nondeterministic problem; it does however, succeed in stabilizing the game, in that: given the strategies of each agent and the locks incurred during play, we could compute the payoff matrix.

It is probably safe to say that if a game is going to be playable it must have a locking mechanism. Otherwise, as mentioned above, there would be no reason to suspect any rational agent would ever take part in it. This concept of playability is not intended to suggest that playing Russian roulette is fun for the whole family. It is most definitely not. It is simply to suggest that the reason the VC took such perverse pleasure in watching Mike pull the trigger is expressly the fact that they knew that he knew he might blow his own head off. If it were not the case that by pulling the trigger on a full chamber guaranteed death, then there would be no reason for the VC to encourage it.

But does this locking mechanism help to fully clarify the DHP? Unfortunately, it does not. We still do not know which type of lock is required. What we need is a more salient view of the terrain and slight detour at this junction may just prove to do the trick. We will now explore the possibility that the synchronization problem can be better described as a game itself. The hope is that this re-expression may help to clarify some of the ideas already presented.

Games as agents ^[6]

This is not as unusual as it may sound, considering games as agents. It has already been established that agents or at least minimal ^[7] agents, as per Viminitz’s description, can be just about anything as long as they satisfy the following criteria: a) some set of representations of some set of states of affairs, coupled with b) at least a partial set of preferences over some of these states of affairs, coupled with c) a set of beliefs about the causal connections between some of these represented states of affairs and some actions available to the agent, coupled with d) an algorithm that fares better than random chance for taking the agent from what it prefers to what it should do to realize that preference, coupled with at least some e) executive capacity. It is easy to see that (a) is satisfied by the DHP itself, but to get the others it will take a little work. Let us now take a look at the configuration of the new game, our meta-game, and address each point in turn.

The question arises as to what type of game this meta-game is. At first glance, it appears as if it should be a non-cooperative one. Two party games, by their nature, are all non-cooperative ^[8]. But, wait. Is it the case in this instance? At second glance, it does not appear to be. It is not as if the agents want a bigger piece of the SRS, it is indivisible anyway. In fact, they do not even care how much access they get to the SRS. All each agent desires is access, and that access is guaranteed. After all, it is clearly their RS we are talking about, it just happens to intersect with that of another game. But then this implies that the agents of this game are indifferent, as they are not concerned with what privileges the other agents are given. But this cannot be correct, defining the meta-game in this manner means that it is not strategic game. What the hell?! This has to be wrong!

As it turns out, the above argument is incorrect; or rather, it must be. While it might be true that the agents of the meta-game are themselves not concerned with access to the SRS the agents of the subordinate games or meta-agents are. This means the agents of the meta-game are merely puppets of the agents of the meta-agent games. For an example of this, let us look at the DHP anew. It can be said that Mike would rather win one of the games than lose any. However, he would rather win the game of Shootout, than the game of Russian roulette. The reason for this is simple, if not obvious; winning the Russian roulette game would inevitable lead to a loss in Shootout, if played, as he would no longer have a partner for it ^[9]. Thus, the desire to win the game of Shootout outweighs that of winning the game of Russian roulette. But this is not the makings of an indifferent agent. It is the makings of a payoff matrix for a very discriminating agent. It follows then, that we have satisfied (b), in that we have shown that a payoff structure can be constructed for this game. It also follows that, (c); the agents of the meta-agent games would prefer one meta-agent to win to another (this concept of priority is discussed next). Finally, the agents of the meta-agent games satisfy (d) and (e).

The priority of the meta-agents is not to suggest that the meta-agent will have a better time of getting access to the SRS, but rather that it will get the last turn. For an example of this, we will look at a few payoff matrices. To derive these matrices we first need to determine the preferences of each agent. These preferences are listed below:

Mike and Nick

1. Mike and Nick prefer to win than lose.
2. Mike and Nick prefer to play Shootout than not.
3. Both prefer to win Shootout than Russian roulette.

Vietcong

1. The Vietcong would prefer to be spectators than participants.
   • Thus, they prefer Mike and Nick to win or lose Russian roulette ^[10].
2. If forced to participate, they would prefer to win.

From these we can construct a set of payoff matrices that will serve as strategies for the meta-agents in the meta-game. For starters, let us take a look that basic game of RR.

Payoff matrix for Mike

	Nick
Mike	Win Lose Win X 1 Lose 2 X

Payoff matrix for Nick

	Nick
Mike	Win Lose Win X 2 Lose 1 X

Table 1: First Russian roulette strategy

The conflict in this game is obvious; each player would rather win and have the other die. This payoff matrix serves as an example of a low priority strategy in the meta-game, as playing it would result in the eventual death of a controlling agent; in this case, either Mike or Nick. This death would conflict with Mike and Nick’s second preference, since the survivor would no longer be in a position to play Shootout. On the bright side, it does conceal their intent of playing Shootout, and surely, that is of benefit.

As it turns out, it is not. Given that both Mike and Nick would rather win the game of Shootout, and that the VC would rather sit it out, we arrive at the following matrix:

Payoff matrix for Mike and Nick

	Vietcong
Mike and Nick	Play ~Play Play 2 1 ~Play X(4)[11] 3

Payoff matrix for Vietcong

	Vietcong
Mike and Nick	Play ~Play Play 2 X(4) ~Play 3 1

Table 2: First Shootout strategy

Thus, neither party can have their cake and eat it too. On the one hand, we have Mike and Nick: they would rather the VC not play Shootout, as it would make their rebellion an instant success. On the other hand, we have the VC, who would rather not participate, but given that Mike and Nick are dead set it, they must. This means everyone is going to play, so Mike and Nick would be better to play Shootout as one of them might die in RR, negating any possibility of playing Shootout in the future. As a result, we get this as a payoff for RR:

Payoff matrix for Mike

	Nick
Mike	Win Lose Win 1 X Lose X X

Payoff matrix for Nick

	Nick
Mike	Win Lose Win 1 X Lose X X

Table 3: Second Russian roulette strategy

This may seem like an unusual payoff, but keep in mind that it reflects the preferences of the agents concerning the meta-game, and not RR.

This leads us in to the most interesting part: if Mike and Nick are going to forgo playing RR, then the force the VC to play Shootout. The result of which lead to a slight change in the original Shootout payoff (Table 2):

Payoff matrix for Mike and Nick

	Vietcong
Mike and Nick	Play ~Play Play 2 1 ~Play 4 3

Payoff matrix for Vietcong

	Vietcong
Mike and Nick	Play ~Play Play 2 4 ~Play 3 1

Table 4: Second Shootout strategy

Notice the addition of a fourth priority; this is because the agents are now committed to playing, thus it is now an option for the VC to initiate Shootout. Also, notice the symmetry of the agents’ moves, it is an EP. Then that means there would be no benefit in playing RR in an attempt to conceal Shootout, as it would only serve to needlessly endanger their lives. What we have shown then, is that there is no true paradox in the DHP; it is simply that the results do not seem rational, when in fact they are.

So there we have it, the re-expression of the synchronization problem in a usable form and an explanation of the mechanics of the DHP. As stated in the opening, this paper was not meant as a solution to the DHP, and I still submit that it is not. Well, not yet, anyway. The reason for this is that there are several problems with the new model that still need to be addressed. For example, it remains to be seen how the representation of the meta-game adds validity and necessity to the locking mechanisms listed above. We also need to establish which locking mechanism is used ^[12]. But, for now, we will let these maters rest.

Footnotes