In 2006, I initially wanted to write my Masters’ Thesis on Dynamic Mixed Equilibrium. Instead, I wrote about the pursuit of higher education. The university was not able to provide an advisor with sufficient knowledge in game theory to guide me in this area. Choosing to write about higher education was the better choice. Developing a theory around a possible Dynamic Mixed Equilibrium is considerably more complex than I could have imagined at that time.
Since graduating, I have revisited my ideas several times. In 2017, I posted a video ‘Game Theory - Mixed Equilibrium’. This video outlined my ideas around reaching a dynamic mixed equilibrium for multi-round and repeat games. The video explored the Nash Mixed Equilibrium as well as my proposed mixed equilibrium. However, the ideas in this video are basic and incomplete. In this post, I elaborate on my ideas in the form of a written post. I believe I am able to offer a considerably more thorough analysis than what I have previously attempted.
A mixed equilibrium occurs when a permanent equilibrium (i.e. all players choose a pure strategy) cannot be determined. This occurs when none of the players have a dominant strategy. A dominant strategy is a strategy that always results in a better outcome for a player regardless of what other players decide to do. Mixed equilibrium only occurs for simultaneous games (i.e. games where all players move at the same time or players are unaware of other players’ moves until the end of the game). In sequential games, an equilibrium can always be determined, as the players select optimal outcomes based on other players’ moves. Figures 1 and 2 contain an example of a sequential game between two players with two choices (i.e. Player One can choose between ‘Up’ or ‘Down’ and Player Two can choose between ‘Left’ and ‘Right’). In this example, the equilibrium differs depending on who moves first.
UL is payoff for ‘Up’ and ‘Left’
UR is payoff for ‘Up’ and ‘Right’
DL is payoff for ‘Down’ and ‘Left’
DR is payoff for ‘Down’ and ‘Right’
Note: The payoffs switch depending on who moves first.
In game theory, we often present payouts in the form of a matrix. Figure 3 shows two 2×2 matrices for a two player game. For this post, I will be discussing all games as two player games where each player has just two options. The same theory can be applied to games with more players with more options but these are more difficult to demonstrate in a post.
In the first matrix, both players have a dominant strategy and in the second matrix, just one player has a dominant strategy. For both these matrices, a permanent equilibrium can be achieved. When both players have a dominant strategy, they will both commit to that strategy. When one player has a dominant strategy, assuming perfect information, the other player will be aware of this dominant strategy. Therefore, the other player will maximise his or her payout based on the other player’s dominant strategy. Hence, equilibrium will be obtained.
In the absence of a dominant strategy for either player, a stable equilibrium cannot be achieved. Instead, the equilibrium is mixed. This means that the same game can reach different results each time it is played. Figure 4 contains an example of a mixed equilibrium.
For the above example, theoretically, all outcomes are possible (i.e. ‘Up-Left’, ‘Up-Right’, ‘Down-Left’, and ‘Down-Right’).
It is common for players not to have dominant strategies. It is also common for games to be repeated. Therefore, it is important to understand how players could or should respond in these games. Below are a few types of games that may not have dominant strategies for any of the players.
I will refer to a game that produces the same equilibrium regardless of who moves first as a game with a preferred outcome. However, sometimes when these games are played simultaneously, neither player has a dominant strategy. For example, developing a particular technology. Companies could benefit more from developing the same technology. This would be apparent in a sequential game but possibly not in a simultaneous game as each other’s actions cannot be observed prior to making a decision.
Battle of sexes is a game that produces better results if players cooperate with each other and/or coordinate activities. In a battle of the sexes game, players benefit from engaging in the same activities more than they benefit from engaging in their preferred activity individually. For example, couples dating. One person may prefer going to the theatre and the other person may prefer going to a sports event but they would both rather go to the theatre or sports event together than attend each event individually. As a sequential game, there is a first mover advantage, as the first mover chooses what they want to do and the second mover would follow for the benefit of the first movers company.
Chicken game is about competition and confrontation between players. Whoever backs down loses but if neither backs down the consequences are worse than choosing to back down. For example, trade talks. It is to the benefit of both parties to reach a deal; however, one party may need to comprise more the other. Comprising may not be the best outcome but it is better than not reaching a deal at all. As a sequential game, there is a first mover advantage. As the first mover sets the terms and conditions of the deal. The second mover either accepts these terms or else a deal will not be reached.
I will refer to a game that produces completely mixed results as a completely mixed game. In this type of game, each player favours a different outcome. In this type of game, there is a second-mover advantage. Mixed results can occur when players have different abilities or offer products of different quality. For example, two firms could offer similar but slightly different quality products. The firms need to decide if they want to invest further in their products to improve their quality. Observing the actions of the other firm would be beneficial to both firms. For example, the firm with the higher quality product may choose to improve their product further if the firm with the lower quality product is attempting to make their product better. If the firm with the lower quality product is not attempting to improve their product, the firm with the higher quality product does not need to improve the quality of their product either.
A zero game is a game where the sum of all payouts always amounts to zero. If a payout is positive for one player, it will be negative for the other player by the equivalent value. For example, a game of American Football. If one team knows what the other team is going to do next, they can offer a stronger response. In reality, American Football is always played as a simultaneous game but circumstances can make certain actions more probable.
A proposed solution to a mixed equilibrium is the Nash mixed equilibrium approach. The Nash mixed equilibrium approach involves calculating the probability that each player will select a particular option. The probability is determined based on what will make the other player indifferent between options. Pursuing indifference produces a mathematically stable outcome (i.e. neither player can improve their position by leaning more to one option). Probability required for indifference is calculated for Player One using the following equation.
UL(P)+ DL(1-P) = UR(P)+ DR(1-P)
(UL-DL)(P) + DL = (UR-DR)(P) + DR
P = (DR - DL)/(UL - DL - UR + DR)
We can apply the above equation for Player One to our example in Figure 4.
7PU + 11(1-PU) = 8PU + 4(1-PU)
-4PU + 11 = 4PU + 4
PU = 7/8
Player One chooses ‘Up’ 7/8 of the time and ‘Down’ 1/8 (1-7/8) of the time.
Probability required for indifference is calculated for Player Two using the following equation.
UL(P)+ UR(1-P) = DL(P)+ DR(1-P)
(UL-UR)(P) + UR = (DL-DR)(P) + DR
P = (DR - UR)/(UL - UR - DL + DR)
We can apply the above equation for Player Two to our example in Figure 4.
12PL + 6(1-PL) = 5PL + 9(1-PL)
6PL + 6 = -4PL + 9
PL = 3/10
Player Two chooses ‘Left’ 3/10 of the time and ‘Right’ 7/10 (1-3/10) of the time.
The chosen strategies can be used to determine the expected payouts for each player. Figure 5 contains the expected payouts for each player.
The combined expected payout for the two players is 15.3. This total is lower than the combined payout of two of the outcomes (i.e. ‘Up-Left’ which has a total of 19 and ‘Down-Left’ which has a total of 16).
I believe the Nash mixed equilibrium more closely resembles a static mixed equilibrium. I would argue this to be the case, as players are not strategizing towards maximising their payoff. Instead, players are working towards preventing the other player/s from gaining an advantage. This approach is logical if the game is played only once and players are allowed to spread their risk by supporting multiple options. In reality, if the game were to be repeated, I would argue that the Nash mixed equilibrium is less likely to occur. If the game is repeated, players have the benefit of observing previous behaviour. This would give strategies greater credibility. In this regard, repeatable simultaneous games are similar to sequential games.
My approach involves creating a dynamic equilibrium, which varies depending on the type of game. Earlier in the post, I discussed five types of games that may not have dominant strategies for any of the players; therefore, the game may require a mixed equilibrium. In this section, I discuss how a dynamic equilibrium could be determined for each of these games.
A game with preferred outcomes may not have any dominant strategies for any players. However, this type of game produces the same outcome in a sequential game regardless of who moves first. Therefore, in a repeat simultaneous game, the same equilibrium that was reached in the sequential games should also be reached in the simultaneous game. Neither player has any incentive to deviate from this outcome, as it is the best possible outcome for both parties. See Figures 6 and 7 below.
A battle of the sexes game does not have dominant strategies for any players nor a consistent outcome if the game was played sequentially (i.e. two possible outcomes depending on who moves first). In the sequential game, players will reach either their most preferred outcome or their second most preferred outcome. In a repeatable simultaneous game, one of these outcomes should prevail as the equilibrium. The outcome will depend on which player has the most to lose from attending an event by themselves. See Figures 8 and 9 below.
In the examples in Figures 8 and 9, Player 2 has the most to lose from the two players parting ways. Therefore, in a repeatable simultaneous game, both players attending the sports event would be the equilibrium.
A chicken game is very similar to a battle of the sexes game in regards to how an equilibrium can be reached. In regards to our earlier chicken game example, we can argue that both players want to reach a deal with minimal compromise. However, someone will need to compromise for a deal to occur. The player who will compromise, will be the player who is hurt the most from not reaching a deal. See Figures 10 and 11 below.
For games such as the preferred outcome, battle of the sexes, and chicken game, I believe it is possible that a single stable equilibrium can be achieved for repeatable simultaneous games. However, for a completely mixed game, a single stable equilibrium is not possible. This is because there is always incentive for at least one player to select an alternative option, which will disrupt an existing outcome. Therefore, at least one player needs to apply a mixed strategy to keep the other player tied to one option. We can analyse the game as if it were a sequential game to determine which outcomes would prevail if each player had the second mover advantage. In Figures 12 and 13 below, we revisit our previous example of two firms deciding whether to upgrade their products or not.
If Firm One had the second mover advantage, the outcome would be that both firms would improve their products. If Firm Two had the second mover advantage, the outcome would be that Firm One would improve their product and Firm Two would not. Both sequential equilibriums involve Firm One improving their product. However, Firm One cannot commit to improving their product as Firm Two would in turn always choose not to improve their product.
Next we should revisit the Nash mixed equilibrium. The Nash mixed equilibrium for this example is for Firm One to upgrade 7/8 of the time and for Firm Two to upgrade 3/10. Firm One has an expected payout of 7.8 and Firm Two has an expected payout of 7.5. If Firm One chooses to upgrade less than 7/8 of the time, Firm Two will always choose to upgrade. If Firm One chooses to upgrade more than 7/8 of the time, Firm Two will always choose to not upgrade. Figure 14 contains the expected payouts under these two scenarios.
Firm One benefits more when Firm Two commits to upgrading than when Firm Two commits to not upgrading. Therefore, Firm One should attempt to entice Firm Two to always upgrade.
If Firm Two chooses to upgrade more than 3/10 of the time, Firm One will always choose to upgrade. If Firm Two chooses to upgrade less than 3/10 of the time, Firm Two will always choose to not upgrade. Figure 15 contains the expected payouts under these two scenarios.
Firm Two benefits more when Firm One commits to upgrading than when Firm One commits to not upgrading. Therefore, Firm Two should attempt to entice Firm One to always upgrade.
Both firms benefit from the other firm’s commitment to upgrade. Which firm is in a better position to entice the other firm to commit to always choosing to upgrade? I believe the answer depends on how much each firm stands to benefit from the other firms commitment. If Firm Two commits to upgrading, Firm One’s expected payout would be slightly less than 11.125. If Firm One commits to upgrading, Firm Two’s expected payout would be slightly less than 7.7.
For Firm Two to obtain a commitment from Firm One to upgrade, Firm Two would need to choose to upgrade a sufficient percent of the time for Firm One to obtain a payout of at least 11.125. The percentage of time Firm Two should choose to upgrade to achieve this can be determined using the formula below.
UUPU2 + UD(1-PU2) = P1ɛ
(UU - UD)PU2 + UD = P1ɛ
PU2 = (P1ɛ - UD)/(UU - UD)
PU2 = (11.125 - 6)/12 - 6) = 85.4167%
UU is an upgrade-upgrade outcome
UD is an upgrade-don’t upgrade outcome
PU2 is the probability of Player Two upgrading
P1ɛ is the maximum payout Firm One can obtain from adopting a Nash mixed equilibrium strategy
For Firm One to obtain a commitment from Firm Two to upgrade, Firm One would need to choose not to upgrade a sufficient percent of the time for Firm Two to obtain a payout of at least 7.7. The percentage of time Firm One should choose to upgrade to achieve this can be determined using the formula below.
UUPU1 + DU(1-PU1) = P2ɛ
(UU - DU)PU1 + DU = P2ɛ
PU1 = (P2ɛ - DU)/(UU - DU)
PU1 = (7.7 - 11)/(7 - 11) = 83%
UU is an upgrade-upgrade outcome
DU is a don’t upgrade-upgrade outcome
PU1 is the probability of Player One upgrading
P2ɛ is the maximum payout Firm One can obtain from adopting a Nash mixed equilibrium strategy
The expected payouts for Firms One and Two have been calculated in Figure 16.
Firm One is able to improve their expected payout as well as offer Firm Two at least the same expected payout under the best case scenario of Firm One always committing to upgrading when Firm Two uses the Nash mixed equilibrium strategy. Firm Two can offer Firm One the same expected payout under the best case scenario of Firm Two always committing to upgrading when Firm One uses the Nash mixed equilibrium strategy but Firm Two’s payout is less than the Nash mixed equilibrium expected payout. Therefore, this strategy would not be pursued by Firm Two. Therefore, in this particular game, a dynamic mixed equilibrium of Firm One upgrading 83% of the time and Firm Two upgrading 100% of the time can be achieved.
The calculated dynamic mixed equilibrium produces higher expected payouts to both Firm One and Firm Two than the Nash mixed equilibrium. This mixed equilibrium is also stable as neither firm has any incentive to change strategy.
A zero sum game is very similar to a completely mixed game. No players have a dominant strategy, sequential games produce different equilibriums depending on who moves first, and there is no possible single equilibrium where neither player has incentive to change strategy. See Figures 17 and 18 for the sequential zero sum game using American Football as an example.
The key difference is that the sum of payouts is the same for each outcome. No outcome is more efficient over another. With a zero sum game, because of the mathematical relationship between payoffs, enticing another player to always commit to the same strategy through a mixed strategy does not affect a player’s expected payoff. Therefore, the Nash mixed equilibrium is the same as the dynamic mixed equilibrium. Figures 19, 20, and 21 uses the American Football zero sum game to demonstrate this relationship.
The dynamic mixed equilibrium offers several possible contributions to game theory; these possible contributions are as follows:
The approach to reaching a dynamic mixed equilibrium varies depending on the types of games we are investigating. In many cases, exploring a possible mixed equilibrium can result in a stable single equilibrium. This was the case, for the preferred option, battle of the sexes and chicken games. The absence of a dominant strategy does not necessarily need to result in a mixed equilibrium for repeated games. As long as an outcome can be reached where none of the players have incentive to adopt an alternative strategy.
For zero sum games and games that are completely mixed, mixed equilibrium is inevitable. For zero sum games, the dynamic mixed equilibrium is the same as the Nash mixed equilibrium as none of the players are able to entice the other players to adopt a pure strategy. For games that are completely mixed, it is possible for one of the players to entice the other players to pursue pure strategies. This can be achieved, if the player’s mixed strategy produces an outcome mutually superior to a Nash mixed equilibrium (i.e. static mixed equilibrium).
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