5.4 Simultaneous Games
 

[5.2 Using Game Theory]    [5.3 Classic Game Models]  

[5.5 Sequential Games]   [5.6 Oligopoly]  

[5.7 Network Effects]  

 

 

 

Simultaneous (or static) games are games in which all players try to make decisions based on what each thinks the others will do and in which no player has any information about the actions of the other players. Each player has to base his or her actions on what he or she thinks (or anticipates) the other player will do.

 

Note that the payoff numbers provided in the various examples presented in this section are arbitrary and only intended to illustrate the concept, not any actual result.

 

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The concept of equilibrium is central to game theory. As stated earlier, equilibrium occurs when each player is using the strategy that will provide them with the optimal payoff given the strategies of all other players. Understanding and employing the idea of equilibrium in games is what allows you to make the right strategic choice for any given situation.

 

An equilibrium strategy provides the highest payoffs to each player given the conditions of that particular game. However, in game theory, terms such as "highest" and "optimal" do not always mean "good". In fact, as in the classic Prisoner's Dilemma game, where equilibrium occurs when both players confess, "optimal" can even mean "not good". Equilibrium strategy simply points you to the best move you can make given the anticipated or actual moves of your opponents.

 

Normal Form Representation of Games

 

A common way of representing games, especially simultaneous games, is the normal form representation, which uses a table structure called a payoff matrix to represent the available strategies (or actions) and the payoffs. For illustrative purposes, suppose that we have two players — Albert and Betty — who are playing a simultaneous game.

 

Albert has the choice of two strategies — moving up or down — while Betty has the choice of left or right. If Albert chooses up and Betty chooses left, Albert receives $20 and Betty receives $20. If Albert and Betty choose up and right, respectively, they receive $20 and $10, respectively. For down and left, the payoffs are $9 and $20. For down and right, the payoffs are $10 and $40. The payoff matrix for this game would look as follows:

 

		Betty
		Left	Right
Albert	Up	$20, $20	$20, $10
	Down	$9, $20	$10, $40

 

This is a conventional method of representing the payoffs with the payoff of the player on the left of the matrix (Albert) listed first and the payoff of the player on the top of the matrix (Betty) listed second.

 

Suppose that we were to allow an additional choice for Albert and Betty so that they could now choose to play middle. The new payoff matrix would look like this:

 

		Betty
		Left	Middle	Right
Albert	Up	$20, $20	$30, $25	$20, $10
	Middle	$11, $25	$9, $20	$30, $10
	Down	$9, $20	$11, $20	$10, $40

 

Thus, if Albert and Betty both play middle, their payoffs are $9 and $20, respectively.

 

Dominant Strategy

 

Once a game has been modelled or represented, as in the above tables, the next thing you want to do is construct a strategy for maximising your payoff, given the variables of the particular game you are playing. Maximising your payoff is different to beating your opponent. In some settings, particular combinations of strategies may lead to higher payoffs for all players than other combinations so doing better than other players may not be consistent with maximising your payoff.

 

 

In simultaneous games, the clearest move a player can make is to follow what game theory refers to as his or her dominant strategy, if one exists. In game theory, rational players can always be counted on to follow their dominant strategy.

 

A dominant strategy is a strategy that provides a player with the highest possible payoff for any strategy of the other players. Note that a dominant strategy does not necessarily provide a player with a higher payoff than the opponents. Being dominant simply means that strategy provides a higher payoff than any other strategy that player could use given the anticipated actions of the opponents.

 

To clarify exactly what this means, consider the following advertising game between competing sellers of bottled water — Kind Lake Bottlers and Red Dolphin Sparkling Water. Suppose that each has to decide between spending a moderate or large amount on advertising. This is a simultaneous game with two players, each of whom has two options. A normal form representation of the game is shown below:

 

		Red Dolphin
		Moderate	Large
Kind Lake	Moderate	$80, $100	$40, $80
	Large	$110, $65	$55, $40

 

Consider the game from the perspective of Kind Lake first. If Red Dolphin is expected to choose a moderate amount of advertising expenditures, Kind Lake's best response is to spend a large amount and earn a profit of $110 (rather than $80). If Red Dolphin is expected to spend a large amount, then Kind Lake's best response is still to spend a large amount and earn $55 (rather than $40). So Kind Lake should spend a large amount regardless of what Red Dolphin does. This means that for Kind Lake, spending a large amount is a dominant strategy — a strategy that provides the highest payoffs no matter what Kind Lake anticipates Red Dolphin will do.

 

 

Now consider the game from the perspective of Red Dolphin. If Red Dolphin's management considers it likely that Kind Lake will spend a moderate amount, Red Dolphin's best response is to also spend a moderate amount and earn $100 (rather than $80). If it is anticipated that Kind Lake will spend a large amount, Red Dolphin's optimal move will again be to spend a moderate amount and earn $65 (rather than $40). So, for Red Dolphin, moderate spending is a dominant strategy. To maximise its payoff, Red Dolphin should spend a moderate amount no matter what it anticipates Kind Lake will do.

 

In this case, Kind Lake choosing a large amount and Red Dolphin choosing a moderate amount is the equilibrium for the game above, as neither grocery would have any desire to switch its choice given the expected choice of the other.

 

Dominated Strategy

 

When a strategy is dominated, it simply means there is at least one other strategy available that offers a higher payoff for each strategy of the other player. A player seeking to maximise payoff should never play a dominated strategy since, by definition, there is always a better or more beneficial strategy available to them in that game. As an example, return to the game between Kind Lake Bottlers and Red Dolphin Sparkling Water.

 

		Red Dolphin
		Moderate	Large
Kind Lake	Moderate	$80, $100	$40, $80
	Large	$110, $65	$55, $40

 

From Kind Lake's perspective, large spending has already been established as the dominant strategy because it offers higher payoffs ($110 or $55) than moderate spending ($80 or $40) no matter what Red Dolphin is expected to do. So for Kind Lake, moderate spending is a dominated strategy — under no circumstances will Kind Lake be better off playing moderate instead of large in this game.

 

Now looking at the table from Red Dolphin's perspective, you see that large is the dominated strategy for similar reasons: under no circumstance will Red Dolphin be better off playing large when it can play moderate — its dominant strategy.

 

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Equilibrium is arrived at differently in sequential and simultaneous games. In some sequential games, players can reach equilibrium by simply responding rationally to an opponent's previous move and anticipated subsequent actions. In simultaneous games, the equilibrium is known as Nash equilibrium and is often harder to attain.

 

Nash equilibrium describes a state in a simultaneous game at which each player has chosen a strategy that optimises his or her payoffs, given the strategies of all other players. What this means is that in a state of Nash equilibrium, no player has an incentive to choose any other strategy because, of all those available, they have chosen the one that provides the optimal payoff — given the anticipated moves of other players. In other words, if you as a player find a game's Nash equilibrium, you find your best move.

 

Consider the earlier example of a game between Albert and Betty:

 

		Betty
		Left	Right
Albert	Up	$20, $20	$20, $10
	Down	$9, $20	$10, $40

 

By studying the payoff matrix, Betty can anticipate that Albert will always go up because, for Albert, up dominates down; that is, regardless of which strategy Betty chooses, Albert's optimal strategy is to play up. So, playing up is a dominant strategy (and playing down is a dominated strategy) for Albert. Believing then that Albert will choose up, which move should Betty make: left or right?

 

Because she anticipates Albert will go up, the Nash equilibrium strategy for Betty is to move left where she will receive a payoff of $20 rather than $10. Here, neither player would have any reason to want to switch his or her move based on the move of the other.

 

		Betty
		Left	Right
Albert	Up	$20, $20	$20, $10
	Down	$9, $20	$10, $40

 

 

 

 

Test your ability to find Nash equilibria in the following exercise. Click here to launch the exercise.

 

Limitations of the Nash Concept

 

Often in simultaneous games, players have more than one Nash equilibrium. In these cases, it can be much harder to narrow down all players' options to just one clear strategy.

 

Suppose, for instance, Albert and Betty are talking on the phone but suddenly get disconnected. If Betty tries to call Albert back, then Albert should stay off the line rather than try to call Betty. However, if Albert decides to wait for Betty to call — but Betty also decides to wait, thinking that Albert will call — the call will never be completed. So the two Nash equilibrium outcomes for this game are Betty calls/Albert waits or Betty waits/Albert calls, as shown in the following payoff matrix:

 

		Betty
		Call	Wait
Albert	Call	0, 0	10, 10
	Wait	10, 10	0, 0

 

Notice that with multiple Nash equilibria neither player has a dominant strategy. This illustrates the limits of the Nash concept: once multiple equilibria have been identified, the concept of Nash equilibrium cannot be used to determine which equilibrium will actually occur — or, in fact, if any will occur.

 

In the real world, communication, even with an opponent, can often be the solution to this problem. However, in situations where communication is not possible, the determination of what outcome (Nash or otherwise) will occur is entirely dependent on the overall conditions of the game.

 

For instance, in the game above, if Albert initiated the first call, it might be logical for him to initiate the second. However, maybe Betty knows Albert's phone number, but he does not know hers, in which case she has to be the one to call. Without clear communication here, the outcome is in question.

 

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Successive Elimination of Dominated Strategies

 

Often a player does not have a clearly dominant strategy — making that game's equilibrium harder to find. In such cases, the next best method a player can use is what game theory refers to as "successive elimination of dominated strategies". This rather lengthy and awkward phrase describes a process by which a player eliminates all strategies that are dominated by some other available strategy that player could adopt.

 

To illustrate this concept, consider once again the advertising game between Kind Lake Bottlers and Red Dolphin Sparkling Flowers. This time, there will be a new series of payoffs.

 

		Red Dolphin
		Moderate	Large
Kind Lake	Moderate	$80, $60	$10, $100
	Large	$50, $10	$40, $30

 

Consider Red Dolphin first. If Red Dolphin anticipates that Kind Lake will choose moderate expenditures, the best action for Red Dolphin is to choose large expenditures and earn $100 (rather than $60). If Kind Lake is expected to choose large expenditures, the best action for Red Dolphin is still to choose large expenditures and earn $30 (rather than $10). Therefore, a large advertising budget is a dominant strategy for Red Dolphin, and moderate is a dominated strategy for Red Dolphin:

 

		Red Dolphin
		Moderate	Large
Kind Lake	Moderate	$80, $60	$10, $100
	Large	$50, $10	$40, $30

 

Now consider Kind Lake. If Kind Lake anticipates Red Dolphin will choose a moderate budget, the best action for Kind Lake is to choose a moderate budget and earn $80 (rather than $50). However, if it is believed that Red Dolphin will choose large expenditures, the best action for Kind Lake is now to choose large and earn $40 (rather than $10). Therefore, the best response for Kind Lake changes and is contingent on the choice made by Red Dolphin. Kind Lake does not, in this case, have a dominant strategy since, as noted above, a dominant strategy has the highest payoff regardless of the moves of a competitor.

 

		Red Dolphin
		Moderate	Large
Kind Lake	Moderate	$80, $60	$10, $100
	Large	$50, $10	$40, $30

 

 

So for Kind Lake, constructing its best strategy, given the payoffs in the above representation, requires it to use successive elimination of dominated strategies.

 

Since Kind Lake's management can see that moderate expenditures is a dominated strategy for Red Dolphin, they can assume that Red Dolphin will not choose moderate no matter what it anticipates Kind Lake will do. Given this, Kind Lake can remove from consideration the option that Red Dolphin will choose moderate. Each time a player eliminates a strategy, the game is changed and prior analysis may no longer apply. Here is the revised matrix with moderate eliminated for Red Dolphin:

 

		Red Dolphin
		Large
Kind Lake	Moderate	$10, $100
	Large	$40, $30

 

 

Knowing now that Red Dolphin's only rational option is to choose large, Kind Lake's best response is to also choose large because its payoff in that case is $40, whereas if it chose moderate its payoff would be only $10. Kind Lake anticipates that Red Dolphin will not play a dominated strategy and, therefore, chooses its optimal action, given that it anticipates Red Dolphin will choose large.

 

		Red Dolphin
		Large
Kind Lake	Moderate	$10, $100
	Large	$40, $30

 

 

The following animations show how players use successive elimination of dominated strategies in a more complex game. Click here to view how Albert eliminates a dominated strategy.

 

 

To continue the elimination of dominated strategies and find the Nash equilibrium of the game, view the animation here.

 

Work through a game theoretic case by clicking on the link here.

 

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The Practicalities of Co-operation

When players communicate, natural barriers preventing those players from achieving higher payoffs can sometimes be removed. Many times, communication simply brings relevant information to light, allowing all players to see the game from the same perspective. Other times, however, communication is intended to bring about co-operation — to get all players to act in a mutually beneficial manner.

But is co-operation ever really practical in a business setting? Will explicit or implicit co-operation between players ever result in the desired outcome? Game theory demonstrates that the answer to these questions is usually — but not always — no.

Suppose that on the hottest day of the year an outdoor festival is held. There are two lemonade vendors at the festival. Knowing that their product will be in high demand, the two vendors agree the night before to make less lemonade than what could be sold. If they restrict their output (reduce the supply), they know prices will go up. Then, if they are able produce less but charge more, both should enjoy considerable profits — if they stick to the agreement. But how likely is it they will stick to the agreement? Will they actually co-operatively restrict their output in order to raise prices, or will they cheat and overproduce? Consider the following payoff matrix:

		Vendor B
		Co-operate	Cheat
Vendor A	Co-operate	$20, $20	$8, $28
	Cheat	$28, $8	$15, $15

 

As you can see, each vendor has a dominant strategy to overproduce. (You may recognise this as a Prisoner's Dilemma game.) Do the payoffs in the matrix above make sense? Consider the case where both vendors co-operate. Here, the vendors are effectively acting as a monopoly and splitting the profits. If both overproduce, the vendors again split the profits. However, this time, they would be in a competitive market and so the overall profits are lower. If one of the vendors deviates, then that vendor will increase profits at the expense of the other vendor.

Clearly, without a binding aspect to the agreement, it will not work in most cases — the incentive to deviate, without enforceable consequence, is usually too great for  any other outcome to occur.

 

Cartels and co-operation: Focus on OPEC

The Organisation of Petroleum Exporting Countries (OPEC) is one of the world's most successful cartels. Since 1960, member countries have met at least twice a year to agree on oil prices and set production quotas. By collectively restricting the amount of oil member countries allow onto the market, OPEC is able to keep oil prices (and profits) artificially high.

 

Without the OPEC agreement, the individual countries would be forced to compete with one another directly in the market, which would cause the prices to come down and the cartel's inflated profits to drop. Knowing this, one could assume that the individual members would be willing to do whatever it takes to keep their cartel healthy.

 

However, in the early 1980s, member countries in need of cash began to deviate from their quotas with regularity. This deviation placed strong downward pressure on the price of oil, which, in turn, forced the non-deviating countries, such as Saudi Arabia, to actually reduce their own output (and therefore, profits) in an attempt to prop prices back up. The deviating countries were essentially stealing from their partners.

 

Adding to the problem of overproduction was the willingness of some member countries, such as heavily indebted Nigeria, to openly undercut OPEC's official, agreed-upon prices whenever the move suited them. These kinds of unauthorised price cuts eventually led to a period of direct competition between cartel members that saw prices fall from $30 a barrel in November of 1985 to a low of only $6 a barrel in July of 1986.

 

OPEC asked the deviators to stop time and time again but, short of declaring war, there was not much the organisation could do to stop its less considerate members, all of whom were sovereign states legally beholden to no one. Had OPEC punished these countries by expelling them from the cartel, it would have lost any hope of controlling them and the cartel's ability to bargain with the rest of the world would have been weakened.

 

Here you can see that even in the world's most "successful" cartel, co-operation between players does not always work. This is because, while it is in the collective interest of all members of the cartel to abide by production quotas and price agreements, it is not in their individual interest to do so. Because all cartel members have an incentive to overproduce or slash prices — absent enforceable threats to punish a cartel member for such behaviour — they will do so whenever the need arises.

OPEC will probably not fall apart any time soon — its members enjoy too many benefits to allow that — but, given human nature, it will also never be the truly co-operative organisation its founders envisioned.

 

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Topic Summary

 

In this topic, you have learnt how to

 

·         formulate simultaneous move games

·         identify dominant strategies

·         find the Nash equilibrium or equilibria of games

·         solve a game by the successive elimination of dominated strategies

·         consider the situations in which co-operation is both desirable and can be achieved

 

Now go on to topic 5.5, “Sequential Games”.