Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training.

The review comment to be evaluated is:

The paper suggests two techniques to improve the calculation of empirically figuring out a Nash equilibrium using an iterative application of best-response dynamics. One method learns the best-response to the previously used strategy. The other uses that technique to model the opponent, and then best-responds to the modeled opponent. The experiments show a faster reaching to NE than without these changes.

The paper is well-written and explained, and is accessible even to researchers not well-versed in ML topics. While, the suggested changes are rather straightforward, they do indeed lead to the expected advance (shorter times to reach NE). I was convinced by further introspection that this is a significant enough contribution, to merit acceptance.

Of course, a more significant change to the algorithm, leading not only to a shorter time but to convergence to better equilibria (in cases where multiple exist) would be far more compelling.

The evidence to be evaluated is:We observe and address two main problems with the standard version of PSRO (Algorithm 1). First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew. Furthermore, because it is responding to some of the same strategies, the new policy learnt may be similar to the strategies added in previous epochs, and so not the most useful addition to the ENFG.

In Section 3.1 we propose an algorithm that transfers knowledge between iterations, and only trains against the single new opponent. In Section 3.2 we present a second algorithm that avoids responding to similar opponent strategies on subsequent iterations, while also addressing the opponent variance issue and providing scalability to multiple other agents.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs.

During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS. A BR to the MSS-generated target mixture is then _constructed_ from the collection of BR results for constituent policies in the mixture.

By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This example motivates our second algorithm: Mixed-Opponents. This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses. The generalized Mixed-Opponent algorithm is shown in Algorithm 3. We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Input: Initial policy sets for all players \(\Pi^{0}\)  Simulate utilities \(\tilde{U}^{\Pi^{0}}\) for each joint \(\pi\!\in\!\Pi^{0}\)  Initialize solutions \(\sigma_{i}^{*,0}\!=\!\text{Uniform}(\Pi_{i}^{0})\) whileepoch e in \(\{1,2,...\}\)do for player \(i\!\in\![[n]]\)do for many episodesdo \(\pi_{-i}\!\sim\!\sigma_{-i}^{*-1}\)  Train \(\pi_{i}^{e}\) over \(\tau\!\sim\!(\pi_{i}^{e}\!,\!\pi_{-i})\) \(\Pi_{i}^{e}\!=\!\Pi_{i}^{-1}\!\cup\!\{\pi_{i}^{e}\}\)  Simulate missing entries in \(\tilde{U}^{\Pi^{e}}\) from \(\Pi^{e}\)  Compute a solution \(\sigma^{*,e}\) from \(\tilde{\Gamma}^{e}\) Output: Current solution \(\sigma_{i}^{*,e}\) for player \(i\)

To address these issues we take a two-step approach (detailed in Section B). First, we select preliminary hyperparameters by searching for their setting that performs best against a random opponent policy. Then we use these hyperparameters to generate a set of opponent-policies (via PSRO) to use as an objective for selecting final hyperparameters. A subset of the generated policies are used as an opponent test-bed to collect two sets of hyperparameters: _pure-hparams_ performed best on-average against each individual test-opponent, and _mix-hparams_ performed best against a uniform mixture of the test-opponents. These settings are meant to represent the high- and low-variance situations encountered when training mixed and pure strategies.

The first research question we address is: does Mixed-Oracles result in a similar quality solution compared to PSRO while utilizing fewer simulation timesteps? We begin by running both algorithms on the Gathering-Small and Leduc-Poker games. We record regrets over PSRO epochs and cumulative training timesteps. Next, we address the question: does Mixed-Opponents result in a better quality solution compared to PSRO while utilizing fewer simulation timesteps?

An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players.

Figure 5(a) shows our results where Mixed-Opponents finds a similar quality solution to PSRO in half of the time.

To answer this question we run all three algorithms with the same set of hyperparameters, forcing all to adopt the same simulation budget.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

In this study, we investigate PSRO's approach to strategy exploration and extensions that focus on reducing the time spent on training new policies.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution.

Next, we introduce Mixed-Opponents which modifies the BR training by constructing a single opponent policy that represents an aggregate of the mixed strategy and learns a response to this new opponent.

Input: Figure 5: PSRO compared against Mixed-Opponents on the Leduc-Poker game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim "We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training" appears to be well-stated. The evidence thoroughly demonstrates both variations (Mixed-Oracles and Mixed-Opponents) and provides experimental results showing reduced simulation requirements while maintaining solution quality. Figure 5 specifically shows Mixed-Opponents finding similar quality solutions in half the time compared to PSRO. The review comment also supports this, noting that the changes lead to "shorter times to reach NE." The claim avoids exaggeration and accurately reflects the demonstrated capabilities.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training.

The review comment to be evaluated is:

The paper focuses on resolving the computational and sample efficiency challenges with current PSRO style approaches. To this end it proposes two different modifications to the standard PSRO setup: 1) Mixed Oracles, and 2) Mixed Opponents. These approaches allow avoiding resetting learning after each outer loop epoch and reduce the stochasticity of dynamics during training. Thee efficacy is demonstrated on relatively simple games but using Deep RL policies where the proposed approaches are at least on par with standard PSRO approach in terms of final performance while drastically improving the sample efficiency.

Although the paper has quite a few typos and unclear writing in places, it has a fantastic setup with clear description of the problems with standard PSRO it's trying to resolve. The paper points out the clear computational deficiencies with current PSRO style approaches: 1) starting anew everytime, 2) slower exploration of strategy space, 3) stochastic dynamics making learning difficult.

The paper is mostly focused on two-player zero sum games.
Approach heavily dependent on efficacy of Q-mixing approach (although the idea is more general), so currently limited to discrete action problems.
Moreover, since there are no alternatives to Q-mixing, we don't actually get any understanding/intuition for why this works and how important is the performance of "TransferOracle" or "OpponentOracle".

Although section 4.4 suggests that the two stage hyperparameter selection wasn't as important, results in Fig 6b are too noisy to fully accept. It's unclear how much compute/samples were required for the hyper parameter selection. In general the results are a lot more noisier for the proposed approaches vs standard PSRO.

**Minor**
- Environment dynamics should be SxA rather than OxA
- Algorithm 4.1 does not exist?
- Fig 1. Is it because of PSRO with Nash. Would a different MSS work differently?
- "proposes MCC as a solution concept": not everyone would know what MCC refers to.

The evidence to be evaluated is:We observe and address two main problems with the standard version of PSRO (Algorithm 1). First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew. Furthermore, because it is responding to some of the same strategies, the new policy learnt may be similar to the strategies added in previous epochs, and so not the most useful addition to the ENFG.

In Section 3.1 we propose an algorithm that transfers knowledge between iterations, and only trains against the single new opponent. In Section 3.2 we present a second algorithm that avoids responding to similar opponent strategies on subsequent iterations, while also addressing the opponent variance issue and providing scalability to multiple other agents.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs.

During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS. A BR to the MSS-generated target mixture is then _constructed_ from the collection of BR results for constituent policies in the mixture.

By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This example motivates our second algorithm: Mixed-Opponents. This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses. The generalized Mixed-Opponent algorithm is shown in Algorithm 3. We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Input: Initial policy sets for all players \(\Pi^{0}\)  Simulate utilities \(\tilde{U}^{\Pi^{0}}\) for each joint \(\pi\!\in\!\Pi^{0}\)  Initialize solutions \(\sigma_{i}^{*,0}\!=\!\text{Uniform}(\Pi_{i}^{0})\) whileepoch e in \(\{1,2,...\}\)do for player \(i\!\in\![[n]]\)do for many episodesdo \(\pi_{-i}\!\sim\!\sigma_{-i}^{*-1}\)  Train \(\pi_{i}^{e}\) over \(\tau\!\sim\!(\pi_{i}^{e}\!,\!\pi_{-i})\) \(\Pi_{i}^{e}\!=\!\Pi_{i}^{-1}\!\cup\!\{\pi_{i}^{e}\}\)  Simulate missing entries in \(\tilde{U}^{\Pi^{e}}\) from \(\Pi^{e}\)  Compute a solution \(\sigma^{*,e}\) from \(\tilde{\Gamma}^{e}\) Output: Current solution \(\sigma_{i}^{*,e}\) for player \(i\)

To address these issues we take a two-step approach (detailed in Section B). First, we select preliminary hyperparameters by searching for their setting that performs best against a random opponent policy. Then we use these hyperparameters to generate a set of opponent-policies (via PSRO) to use as an objective for selecting final hyperparameters. A subset of the generated policies are used as an opponent test-bed to collect two sets of hyperparameters: _pure-hparams_ performed best on-average against each individual test-opponent, and _mix-hparams_ performed best against a uniform mixture of the test-opponents. These settings are meant to represent the high- and low-variance situations encountered when training mixed and pure strategies.

The first research question we address is: does Mixed-Oracles result in a similar quality solution compared to PSRO while utilizing fewer simulation timesteps? We begin by running both algorithms on the Gathering-Small and Leduc-Poker games. We record regrets over PSRO epochs and cumulative training timesteps. Next, we address the question: does Mixed-Opponents result in a better quality solution compared to PSRO while utilizing fewer simulation timesteps?

An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players.

Figure 5(a) shows our results where Mixed-Opponents finds a similar quality solution to PSRO in half of the time.

To answer this question we run all three algorithms with the same set of hyperparameters, forcing all to adopt the same simulation budget.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

In this study, we investigate PSRO's approach to strategy exploration and extensions that focus on reducing the time spent on training new policies.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution.

Next, we introduce Mixed-Opponents which modifies the BR training by constructing a single opponent policy that represents an aggregate of the mixed strategy and learns a response to this new opponent.

Input: Figure 5: PSRO compared against Mixed-Opponents on the Leduc-Poker game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Based on my review of the evidence and reviewer comments, I'll assess whether the claim "We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training" is well-stated or overstated.

Summary:
The evidence strongly supports this claim. The paper clearly demonstrates both variations (Mixed-Oracles and Mixed-Opponents) and provides experimental results showing reduced simulation requirements. Figure 5 specifically shows Mixed-Opponents achieving similar quality solutions in half the timesteps. The methods are well-explained with their theoretical foundations, and limitations are acknowledged. While there are some limitations noted in the review (like noise in results and dependency on Q-mixing), these don't invalidate the core claim about reducing simulation requirements.

<score>1</score>

Justification:
1. The paper provides clear theoretical foundations for both variations and explains how they reduce simulation requirements through:
- Mixed-Oracles: Transferring knowledge between iterations and training only against new opponents
- Mixed-Opponents: Combining opponent strategies to create a single training target

2. Experimental evidence directly supports the claim:
- Figure 5 shows Mixed-Opponents achieving similar results in half the timesteps
- Results demonstrate that "Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps"

3. The authors acknowledge limitations:
- Focus on two-player zero-sum games
- Dependency on Q-mixing approach
- Applicability to discrete action problems

4. The reviewer's comments about noise in results and hyperparameter selection don't contradict the core claim about reducing simulation requirements.

The claim is well-stated (score of 1) because it accurately reflects what the paper demonstrates without overstating the findings or ignoring limitations.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training.

The review comment to be evaluated is:

Summary
The paper proposes two new methods in the Policy-Space Response Oracle framework. These approaches permit to reuse past knowledge in order to reduce the amount of data required for the RL training. The first algorithm Mixed-Oracles transfers the previous iteration of Deep RL, instead of the second one, Mixed-Opponents, transfers existing strategy action-value estimates.

Strengths
The paper proposes two convincing alternatives to reuse previous knowledge in the PSRO framework. The two ideas are based on Q-mixing approach: the first one uses this method to transfer Q-values across epochs, the second one to design a new training objective.
The experiments show that the proposed methods find a good solution using less simulation than the original PSRO framework.

Weakness
The paper is not very novel, since it uses previous approaches (PSRO and Q-mixing) to transfer knowledge for the PSRO framework.
I am not aware of recent works in this framework but could be useful to compare the proposed approaches with P2RO [1].
The Mixed-Opponent section needs a better explanation of the use of Q-mixing as a training objective.

[1]  McAleer, S., Lanier, J., Fox, R., & Baldi, P. (2020). Pipeline PSRO: A Scalable Approach for Finding Approximate Nash Equilibria in Large Games. arXiv preprint arXiv:2006.08555.

The evidence to be evaluated is:We observe and address two main problems with the standard version of PSRO (Algorithm 1). First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew. Furthermore, because it is responding to some of the same strategies, the new policy learnt may be similar to the strategies added in previous epochs, and so not the most useful addition to the ENFG.

In Section 3.1 we propose an algorithm that transfers knowledge between iterations, and only trains against the single new opponent. In Section 3.2 we present a second algorithm that avoids responding to similar opponent strategies on subsequent iterations, while also addressing the opponent variance issue and providing scalability to multiple other agents.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs.

During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS. A BR to the MSS-generated target mixture is then _constructed_ from the collection of BR results for constituent policies in the mixture.

By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This example motivates our second algorithm: Mixed-Opponents. This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses. The generalized Mixed-Opponent algorithm is shown in Algorithm 3. We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Input: Initial policy sets for all players \(\Pi^{0}\)  Simulate utilities \(\tilde{U}^{\Pi^{0}}\) for each joint \(\pi\!\in\!\Pi^{0}\)  Initialize solutions \(\sigma_{i}^{*,0}\!=\!\text{Uniform}(\Pi_{i}^{0})\) whileepoch e in \(\{1,2,...\}\)do for player \(i\!\in\![[n]]\)do for many episodesdo \(\pi_{-i}\!\sim\!\sigma_{-i}^{*-1}\)  Train \(\pi_{i}^{e}\) over \(\tau\!\sim\!(\pi_{i}^{e}\!,\!\pi_{-i})\) \(\Pi_{i}^{e}\!=\!\Pi_{i}^{-1}\!\cup\!\{\pi_{i}^{e}\}\)  Simulate missing entries in \(\tilde{U}^{\Pi^{e}}\) from \(\Pi^{e}\)  Compute a solution \(\sigma^{*,e}\) from \(\tilde{\Gamma}^{e}\) Output: Current solution \(\sigma_{i}^{*,e}\) for player \(i\)

To address these issues we take a two-step approach (detailed in Section B). First, we select preliminary hyperparameters by searching for their setting that performs best against a random opponent policy. Then we use these hyperparameters to generate a set of opponent-policies (via PSRO) to use as an objective for selecting final hyperparameters. A subset of the generated policies are used as an opponent test-bed to collect two sets of hyperparameters: _pure-hparams_ performed best on-average against each individual test-opponent, and _mix-hparams_ performed best against a uniform mixture of the test-opponents. These settings are meant to represent the high- and low-variance situations encountered when training mixed and pure strategies.

The first research question we address is: does Mixed-Oracles result in a similar quality solution compared to PSRO while utilizing fewer simulation timesteps? We begin by running both algorithms on the Gathering-Small and Leduc-Poker games. We record regrets over PSRO epochs and cumulative training timesteps. Next, we address the question: does Mixed-Opponents result in a better quality solution compared to PSRO while utilizing fewer simulation timesteps?

An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players.

Figure 5(a) shows our results where Mixed-Opponents finds a similar quality solution to PSRO in half of the time.

To answer this question we run all three algorithms with the same set of hyperparameters, forcing all to adopt the same simulation budget.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

In this study, we investigate PSRO's approach to strategy exploration and extensions that focus on reducing the time spent on training new policies.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution.

Next, we introduce Mixed-Opponents which modifies the BR training by constructing a single opponent policy that represents an aggregate of the mixed strategy and learns a response to this new opponent.

Input: Figure 5: PSRO compared against Mixed-Opponents on the Leduc-Poker game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Based on the evidence provided, the claim "We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training" appears to be well-stated and accurately reflects the research findings.

The evidence demonstrates that:
1. Both Mixed-Oracles and Mixed-Opponents algorithms achieve comparable or better solutions while using fewer simulation timesteps than standard PSRO
2. Figure 5 specifically shows Mixed-Opponents finding similar quality solutions in half the time
3. The experimental results validate the claim's core assertion about reducing simulation requirements
4. The review comment confirms that "experiments show that the proposed methods find a good solution using less simulation than the original PSRO framework"

The claim is modest in scope and directly supported by the evidence without exaggeration.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training.

The review comment to be evaluated is:

##########################################################################
Summary:

The paper provides an interesting approach to speeding up the convergence time of the Policy-Space Response Oracles framework by re-using the Q-functions of past best-responses to transfer knowledge across epochs.
##########################################################################
Reasons for score:

Overall, I am low confidence on my assessment of this paper due to the exposition in the algorithm section being relatively confusing. The experimental results are interesting which suggests that the method has value but there is key missing information on how the best response policies are constructed that make it difficult to assess the paper and lead to my not wanting to recommend its acceptance. I would highly recommend being more detailed in Sec. 3 to allow me to reassess the paper. I would certainly be willing to update my score if the paper was clearer to read.
 ##########################################################################Pros:

1. The experimental results on Leduc Poker are very speedy in terms of time-steps.

2. The idea of reusing the prior Q functions and just mixing them together rather than re-learning all of the policies is very good.

##########################################################################
Cons:

1. The algorithm boxes are so high level that I am struggling to understand how the algorithms work. I would not be able to implement it from reading the paper.

#########################################################################
Things that would improve readability:

- It would be nice in the algorithm boxes to connect Q-mixing to how the best policy is explicitly output. I was not able to understand how Q-mixing was connected to either Algorithm 2 or 3 and subsequently had difficulty following the paper.
- \lambda does not appear to be defined anywhere but appears in the Mixed Oracles algorithm box
- What is the OpponentOracle and the TransferOracle? They are defined in the algorithm boxes but are not clearly defined elsewhere.
- The specific example of RPS in section 3.2 does not provide useful intuition by going through the numerics, it may be more helpful to walk through a more high level description.
- It would probably be useful to move more of the experimental results to an appendix to leave room for the exposition of the algorithms.

The evidence to be evaluated is:We observe and address two main problems with the standard version of PSRO (Algorithm 1). First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew. Furthermore, because it is responding to some of the same strategies, the new policy learnt may be similar to the strategies added in previous epochs, and so not the most useful addition to the ENFG.

In Section 3.1 we propose an algorithm that transfers knowledge between iterations, and only trains against the single new opponent. In Section 3.2 we present a second algorithm that avoids responding to similar opponent strategies on subsequent iterations, while also addressing the opponent variance issue and providing scalability to multiple other agents.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs.

During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS. A BR to the MSS-generated target mixture is then _constructed_ from the collection of BR results for constituent policies in the mixture.

By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This example motivates our second algorithm: Mixed-Opponents. This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses. The generalized Mixed-Opponent algorithm is shown in Algorithm 3. We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Input: Initial policy sets for all players \(\Pi^{0}\)  Simulate utilities \(\tilde{U}^{\Pi^{0}}\) for each joint \(\pi\!\in\!\Pi^{0}\)  Initialize solutions \(\sigma_{i}^{*,0}\!=\!\text{Uniform}(\Pi_{i}^{0})\) whileepoch e in \(\{1,2,...\}\)do for player \(i\!\in\![[n]]\)do for many episodesdo \(\pi_{-i}\!\sim\!\sigma_{-i}^{*-1}\)  Train \(\pi_{i}^{e}\) over \(\tau\!\sim\!(\pi_{i}^{e}\!,\!\pi_{-i})\) \(\Pi_{i}^{e}\!=\!\Pi_{i}^{-1}\!\cup\!\{\pi_{i}^{e}\}\)  Simulate missing entries in \(\tilde{U}^{\Pi^{e}}\) from \(\Pi^{e}\)  Compute a solution \(\sigma^{*,e}\) from \(\tilde{\Gamma}^{e}\) Output: Current solution \(\sigma_{i}^{*,e}\) for player \(i\)

To address these issues we take a two-step approach (detailed in Section B). First, we select preliminary hyperparameters by searching for their setting that performs best against a random opponent policy. Then we use these hyperparameters to generate a set of opponent-policies (via PSRO) to use as an objective for selecting final hyperparameters. A subset of the generated policies are used as an opponent test-bed to collect two sets of hyperparameters: _pure-hparams_ performed best on-average against each individual test-opponent, and _mix-hparams_ performed best against a uniform mixture of the test-opponents. These settings are meant to represent the high- and low-variance situations encountered when training mixed and pure strategies.

The first research question we address is: does Mixed-Oracles result in a similar quality solution compared to PSRO while utilizing fewer simulation timesteps? We begin by running both algorithms on the Gathering-Small and Leduc-Poker games. We record regrets over PSRO epochs and cumulative training timesteps. Next, we address the question: does Mixed-Opponents result in a better quality solution compared to PSRO while utilizing fewer simulation timesteps?

An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players.

Figure 5(a) shows our results where Mixed-Opponents finds a similar quality solution to PSRO in half of the time.

To answer this question we run all three algorithms with the same set of hyperparameters, forcing all to adopt the same simulation budget.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

In this study, we investigate PSRO's approach to strategy exploration and extensions that focus on reducing the time spent on training new policies.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution.

Next, we introduce Mixed-Opponents which modifies the BR training by constructing a single opponent policy that represents an aggregate of the mixed strategy and learns a response to this new opponent.

Input: Figure 5: PSRO compared against Mixed-Opponents on the Leduc-Poker game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Based on the evidence provided, I'll evaluate the claim "We introduce two variations of PSRO designed to reduce the amount of simulation required during Deep RL training."

The evidence strongly supports this claim through:
1. Detailed explanations of both variations (Mixed-Oracles and Mixed-Opponents)
2. Clear experimental results showing reduced simulation requirements (Figure 5 demonstrates Mixed-Opponents finding similar quality solutions in half the timesteps)
3. Specific mechanics of how each variation reduces training time (reusing learned behaviors, combining Q-values)
4. Review comments confirming "very speedy" results on Leduc Poker

The claim is well-stated and accurately reflects the evidence without exaggeration. The methods are clearly described and supported by experimental results.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oracles transfers learning across epochs, exploiting the Q-functions learned against a particular opponent policy in constructing policies for any other epoch where that opponent policy is encountered.

The evidence to be evaluated is:At each epoch \(e\) of the PSRO algorithm a new policy is constructed for each player by best-responding to an opponent profile \(\sigma_{-i}^{*,e-1}\) from the currently constructed ENFG: \(\pi_{i}^{e}\in\text{BR}(\sigma_{-i}^{*,e-1})\). These policies are then added to each player's strategy set, \(\tilde{\Pi}_{i}^{e}\!\leftarrow\!\tilde{\Pi}_{i}^{e-1}\!\cup\!\left\{\pi_{i}^{ e}\right\}\), and the new profiles are simulated to expand the ENFG.

First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs. This method works by learning and maintaining a collection of BRs to each opponent policy \(\Lambda_{i}^{e}\!=\!\left\{\lambda_{1}^{1},\!...,\!\lambda_{i}^{e}\right\}\), where \(\lambda_{i}^{e}\) is the BR to \(\pi_{-i}^{e-1}\). During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS.

Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies. By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This can be detected by inspecting player 2's Q-functions: \(Q_{2}^{1}\) has a very low valuation of P, and \(Q_{2}^{2}\) has a very low valuation of R.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses.

We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution. This affords our algorithm the ability to transfer learning about previously encountered opponents.

Input: Figure 2: PSRO compared against Mixed-Oracles on the Gathering-Small game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, the Mixed-Oracles algorithm does demonstrate transfer learning across epochs through Q-Mixing, which combines Q-values learned against individual opponent policies weighted by their likelihood in the opponent mixture. The evidence shows both theoretical foundation (Equation 1) and empirical results (Figure 2) where Mixed-Oracles achieves better performance in fewer timesteps compared to standard PSRO. The claim is well-stated as it accurately reflects the mechanism of transfer learning without exaggerating its capabilities or implications.

<score>1</score>

Justification:
1. The evidence directly supports the claim through:
- Mathematical formulation showing how Q-values are transferred (Equation 1)
- Explicit description of maintaining BRs to each opponent policy (Λᵢᵉ)
- Empirical results in Figure 2 demonstrating faster convergence
2. No exaggeration is present - the claim specifically states it "exploits Q-functions learned against a particular opponent policy" which is precisely what the evidence shows
3. The mechanism of transfer is clearly explained in the evidence through Q-Mixing
4. Results validate the transfer learning benefit with improved performance in fewer timesteps


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oracles transfers learning across epochs, exploiting the Q-functions learned against a particular opponent policy in constructing policies for any other epoch where that opponent policy is encountered.

The review comment to be evaluated is:

The paper suggests two techniques to improve the calculation of empirically figuring out a Nash equilibrium using an iterative application of best-response dynamics. One method learns the best-response to the previously used strategy. The other uses that technique to model the opponent, and then best-responds to the modeled opponent. The experiments show a faster reaching to NE than without these changes.

The paper is well-written and explained, and is accessible even to researchers not well-versed in ML topics. While, the suggested changes are rather straightforward, they do indeed lead to the expected advance (shorter times to reach NE). I was convinced by further introspection that this is a significant enough contribution, to merit acceptance.

Of course, a more significant change to the algorithm, leading not only to a shorter time but to convergence to better equilibria (in cases where multiple exist) would be far more compelling.

The evidence to be evaluated is:At each epoch \(e\) of the PSRO algorithm a new policy is constructed for each player by best-responding to an opponent profile \(\sigma_{-i}^{*,e-1}\) from the currently constructed ENFG: \(\pi_{i}^{e}\in\text{BR}(\sigma_{-i}^{*,e-1})\). These policies are then added to each player's strategy set, \(\tilde{\Pi}_{i}^{e}\!\leftarrow\!\tilde{\Pi}_{i}^{e-1}\!\cup\!\left\{\pi_{i}^{ e}\right\}\), and the new profiles are simulated to expand the ENFG.

First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs. This method works by learning and maintaining a collection of BRs to each opponent policy \(\Lambda_{i}^{e}\!=\!\left\{\lambda_{1}^{1},\!...,\!\lambda_{i}^{e}\right\}\), where \(\lambda_{i}^{e}\) is the BR to \(\pi_{-i}^{e-1}\). During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS.

Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies. By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This can be detected by inspecting player 2's Q-functions: \(Q_{2}^{1}\) has a very low valuation of P, and \(Q_{2}^{2}\) has a very low valuation of R.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses.

We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution. This affords our algorithm the ability to transfer learning about previously encountered opponents.

Input: Figure 2: PSRO compared against Mixed-Oracles on the Gathering-Small game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find that the claim is reasonably well-stated but has some minor overstatement. The evidence shows that Mixed-Oracles does transfer learning by maintaining a collection of best responses to each opponent policy and using Q-Mixing to combine these responses. The graphs demonstrate improved performance over PSRO, both in terms of epochs and timesteps. However, the claim could be more precise about the implementation details and limitations of this transfer learning approach.

The algorithm transfers learning primarily through Q-Mixing of previously learned Q-functions, rather than broadly "exploiting Q-functions" as suggested in the claim. While the core statement is supported by the evidence, the wording implies a slightly more general capability than what is specifically demonstrated.

<score>2</score>

Justification:
1. The evidence directly supports the transfer learning concept through the mathematical formulation of Q-Mixing (Equation 1) and the description of maintaining best responses Λᵢᵉ.
2. The performance graphs in Figure 2 demonstrate the effectiveness of this transfer learning approach.
3. Review comments confirm the practical improvements but also note the straightforward nature of the changes.
4. The claim could be more precise about using Q-Mixing specifically as the transfer mechanism rather than implying general Q-function exploitation.
5. The evidence shows the transfer works specifically for opponent policies in mixtures, rather than any arbitrary epoch encounter as the claim might suggest.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oracles transfers learning across epochs, exploiting the Q-functions learned against a particular opponent policy in constructing policies for any other epoch where that opponent policy is encountered.

The review comment to be evaluated is:

The paper focuses on resolving the computational and sample efficiency challenges with current PSRO style approaches. To this end it proposes two different modifications to the standard PSRO setup: 1) Mixed Oracles, and 2) Mixed Opponents. These approaches allow avoiding resetting learning after each outer loop epoch and reduce the stochasticity of dynamics during training. Thee efficacy is demonstrated on relatively simple games but using Deep RL policies where the proposed approaches are at least on par with standard PSRO approach in terms of final performance while drastically improving the sample efficiency.

Although the paper has quite a few typos and unclear writing in places, it has a fantastic setup with clear description of the problems with standard PSRO it's trying to resolve. The paper points out the clear computational deficiencies with current PSRO style approaches: 1) starting anew everytime, 2) slower exploration of strategy space, 3) stochastic dynamics making learning difficult.

The paper is mostly focused on two-player zero sum games.
Approach heavily dependent on efficacy of Q-mixing approach (although the idea is more general), so currently limited to discrete action problems.
Moreover, since there are no alternatives to Q-mixing, we don't actually get any understanding/intuition for why this works and how important is the performance of "TransferOracle" or "OpponentOracle".

Although section 4.4 suggests that the two stage hyperparameter selection wasn't as important, results in Fig 6b are too noisy to fully accept. It's unclear how much compute/samples were required for the hyper parameter selection. In general the results are a lot more noisier for the proposed approaches vs standard PSRO.

**Minor**
- Environment dynamics should be SxA rather than OxA
- Algorithm 4.1 does not exist?
- Fig 1. Is it because of PSRO with Nash. Would a different MSS work differently?
- "proposes MCC as a solution concept": not everyone would know what MCC refers to.

The evidence to be evaluated is:At each epoch \(e\) of the PSRO algorithm a new policy is constructed for each player by best-responding to an opponent profile \(\sigma_{-i}^{*,e-1}\) from the currently constructed ENFG: \(\pi_{i}^{e}\in\text{BR}(\sigma_{-i}^{*,e-1})\). These policies are then added to each player's strategy set, \(\tilde{\Pi}_{i}^{e}\!\leftarrow\!\tilde{\Pi}_{i}^{e-1}\!\cup\!\left\{\pi_{i}^{ e}\right\}\), and the new profiles are simulated to expand the ENFG.

First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs. This method works by learning and maintaining a collection of BRs to each opponent policy \(\Lambda_{i}^{e}\!=\!\left\{\lambda_{1}^{1},\!...,\!\lambda_{i}^{e}\right\}\), where \(\lambda_{i}^{e}\) is the BR to \(\pi_{-i}^{e-1}\). During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS.

Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies. By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This can be detected by inspecting player 2's Q-functions: \(Q_{2}^{1}\) has a very low valuation of P, and \(Q_{2}^{2}\) has a very low valuation of R.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses.

We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution. This affords our algorithm the ability to transfer learning about previously encountered opponents.

Input: Figure 2: PSRO compared against Mixed-Oracles on the Gathering-Small game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll analyze the claim about Mixed-Oracles' transfer learning capabilities across epochs.

The evidence strongly supports this claim through:
1. Clear theoretical explanation of how Mixed-Oracles maintains and reuses BRs to each opponent policy (Λᵢᵉ)
2. Explicit mathematical formulation showing Q-Mixing implementation for transfer
3. Figure 2 demonstrates practical effectiveness, showing Mixed-Oracles achieving better performance in fewer timesteps
4. The reviewer comment acknowledges the sample efficiency improvements

The claim is well-stated (score 1) as it accurately reflects the mechanism without exaggeration - it specifically mentions "exploiting Q-functions" and limits the scope to "where that opponent policy is encountered," showing appropriate restraint in the claim's scope.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oracles transfers learning across epochs, exploiting the Q-functions learned against a particular opponent policy in constructing policies for any other epoch where that opponent policy is encountered.

The review comment to be evaluated is:

Summary
The paper proposes two new methods in the Policy-Space Response Oracle framework. These approaches permit to reuse past knowledge in order to reduce the amount of data required for the RL training. The first algorithm Mixed-Oracles transfers the previous iteration of Deep RL, instead of the second one, Mixed-Opponents, transfers existing strategy action-value estimates.

Strengths
The paper proposes two convincing alternatives to reuse previous knowledge in the PSRO framework. The two ideas are based on Q-mixing approach: the first one uses this method to transfer Q-values across epochs, the second one to design a new training objective.
The experiments show that the proposed methods find a good solution using less simulation than the original PSRO framework.

Weakness
The paper is not very novel, since it uses previous approaches (PSRO and Q-mixing) to transfer knowledge for the PSRO framework.
I am not aware of recent works in this framework but could be useful to compare the proposed approaches with P2RO [1].
The Mixed-Opponent section needs a better explanation of the use of Q-mixing as a training objective.

[1]  McAleer, S., Lanier, J., Fox, R., & Baldi, P. (2020). Pipeline PSRO: A Scalable Approach for Finding Approximate Nash Equilibria in Large Games. arXiv preprint arXiv:2006.08555.

The evidence to be evaluated is:At each epoch \(e\) of the PSRO algorithm a new policy is constructed for each player by best-responding to an opponent profile \(\sigma_{-i}^{*,e-1}\) from the currently constructed ENFG: \(\pi_{i}^{e}\in\text{BR}(\sigma_{-i}^{*,e-1})\). These policies are then added to each player's strategy set, \(\tilde{\Pi}_{i}^{e}\!\leftarrow\!\tilde{\Pi}_{i}^{e-1}\!\cup\!\left\{\pi_{i}^{ e}\right\}\), and the new profiles are simulated to expand the ENFG.

First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs. This method works by learning and maintaining a collection of BRs to each opponent policy \(\Lambda_{i}^{e}\!=\!\left\{\lambda_{1}^{1},\!...,\!\lambda_{i}^{e}\right\}\), where \(\lambda_{i}^{e}\) is the BR to \(\pi_{-i}^{e-1}\). During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS.

Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies. By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This can be detected by inspecting player 2's Q-functions: \(Q_{2}^{1}\) has a very low valuation of P, and \(Q_{2}^{2}\) has a very low valuation of R.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses.

We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution. This affords our algorithm the ability to transfer learning about previously encountered opponents.

Input: Figure 2: PSRO compared against Mixed-Oracles on the Gathering-Small game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, the claim about Mixed-Oracles transferring learning across epochs appears well-supported. The evidence clearly describes how Mixed-Oracles maintains a collection of best responses to each opponent policy and reuses them across epochs through Q-Mixing. The mathematical formulation (Equation 1) and implementation details are provided. Figure 2 demonstrates empirical validation, showing Mixed-Oracles achieving better performance with fewer timesteps compared to standard PSRO. The reviewer's comments also acknowledge the method's effectiveness in reusing previous knowledge. The claim accurately reflects the evidence without exaggeration.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oracles transfers learning across epochs, exploiting the Q-functions learned against a particular opponent policy in constructing policies for any other epoch where that opponent policy is encountered.

The review comment to be evaluated is:

##########################################################################
Summary:

The paper provides an interesting approach to speeding up the convergence time of the Policy-Space Response Oracles framework by re-using the Q-functions of past best-responses to transfer knowledge across epochs.
##########################################################################
Reasons for score:

Overall, I am low confidence on my assessment of this paper due to the exposition in the algorithm section being relatively confusing. The experimental results are interesting which suggests that the method has value but there is key missing information on how the best response policies are constructed that make it difficult to assess the paper and lead to my not wanting to recommend its acceptance. I would highly recommend being more detailed in Sec. 3 to allow me to reassess the paper. I would certainly be willing to update my score if the paper was clearer to read.
 ##########################################################################Pros:

1. The experimental results on Leduc Poker are very speedy in terms of time-steps.

2. The idea of reusing the prior Q functions and just mixing them together rather than re-learning all of the policies is very good.

##########################################################################
Cons:

1. The algorithm boxes are so high level that I am struggling to understand how the algorithms work. I would not be able to implement it from reading the paper.

#########################################################################
Things that would improve readability:

- It would be nice in the algorithm boxes to connect Q-mixing to how the best policy is explicitly output. I was not able to understand how Q-mixing was connected to either Algorithm 2 or 3 and subsequently had difficulty following the paper.
- \lambda does not appear to be defined anywhere but appears in the Mixed Oracles algorithm box
- What is the OpponentOracle and the TransferOracle? They are defined in the algorithm boxes but are not clearly defined elsewhere.
- The specific example of RPS in section 3.2 does not provide useful intuition by going through the numerics, it may be more helpful to walk through a more high level description.
- It would probably be useful to move more of the experimental results to an appendix to leave room for the exposition of the algorithms.

The evidence to be evaluated is:At each epoch \(e\) of the PSRO algorithm a new policy is constructed for each player by best-responding to an opponent profile \(\sigma_{-i}^{*,e-1}\) from the currently constructed ENFG: \(\pi_{i}^{e}\in\text{BR}(\sigma_{-i}^{*,e-1})\). These policies are then added to each player's strategy set, \(\tilde{\Pi}_{i}^{e}\!\leftarrow\!\tilde{\Pi}_{i}^{e-1}\!\cup\!\left\{\pi_{i}^{ e}\right\}\), and the new profiles are simulated to expand the ENFG.

First, note that the only thing that changes from one epoch to the next is the other-agent strategy profile \(\sigma_{-i}\); the transition dynamics remain the same, and the mixed profile \(\sigma_{-i}\) itself likely contains other-agent policies encountered in previous epochs. This suggests a compelling opportunity to transfer learning across epochs; however, the BR calculation step works by training anew.

The _Mixed-Oracles_ algorithm is a variant of PSRO for two-player games, with a modified BR oracle designed to transfer learning across epochs. This method works by learning and maintaining a collection of BRs to each opponent policy \(\Lambda_{i}^{e}\!=\!\left\{\lambda_{1}^{1},\!...,\!\lambda_{i}^{e}\right\}\), where \(\lambda_{i}^{e}\) is the BR to \(\pi_{-i}^{e-1}\). During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS.

Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies. By reusing learned behaviors from previous epochs, Mixed-Oracles allows us to focus training exclusively on new opponent policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This can be detected by inspecting player 2's Q-functions: \(Q_{2}^{1}\) has a very low valuation of P, and \(Q_{2}^{2}\) has a very low valuation of R.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training. We refer to the method for generating a new opponent policy from a mixture of opponents the Combine-Oponents, and it has the same functional form as the Combine-Responses.

We employ Q-Mixing (Equation 1) as our Combine-Oponents. In contrast to Mixed-Oracles, which uses Q-Mixing to transfer Q-values across epochs, here we apply it to average Q-values to define a variant training objective.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

We first propose Mixed-Oracles which approximates the solution step by training separate BRs to each opponent policy and then combining the results into an approximate BR to the solution. This affords our algorithm the ability to transfer learning about previously encountered opponents.

Input: Figure 2: PSRO compared against Mixed-Oracles on the Gathering-Small game. SumRegret over ΠPSRO∪ΠEVAL is compared over PSRO epochs (left) and training timesteps (right).

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find that it is generally well-stated but has some minor overstatement. The evidence confirms that Mixed-Oracles does transfer learning across epochs by reusing Q-functions from previous best responses, as demonstrated through Q-Mixing equations and experimental results (Figure 2 shows improved performance). However, the claim could be more precise about the mechanism - it's specifically through Q-Mixing of previously learned Q-functions, not direct policy transfer. The review comments also note some lack of clarity in the algorithm details, though they don't directly contradict the claim's core assertion.

<score>2</score>

Justification:
1. Supporting evidence:
- The text explicitly describes the Q-Mixing mechanism for transfer
- Figure 2 demonstrates improved performance
- The mathematical formulation in Equation 1 shows how Q-values are combined

2. Minor overstatement concerns:
- The claim suggests universal applicability ("any other epoch") which isn't fully demonstrated
- The review points out some unclear algorithm details
- The mechanism is more specific (Q-function mixing) than the broad "transfers learning" statement

3. The claim captures the core concept accurately but could be more precise in its scope and mechanism description.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oppennts applies directly over the joint opponent space, and so has the potential to scale beyond two-player games.

The evidence to be evaluated is:A variety of other MSSs have been proposed, and assessing their relative merits is a topic of active research (Balduzzi et al., 2019). In experiments reported here, we adopt NE as the baseline MSS; however, our methods readily apply to any MSS.

In Section 3.2 we present a second algorithm that avoids responding to similar opponent strategies on subsequent iterations, while also addressing the opponent variance issue and providing scalability to multiple other agents.

During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS. A BR to the MSS-generated target mixture is then _constructed_ from the collection of BR results for constituent policies in the mixture. Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies.

The key design choice is how to combine knowledge from the BRs to individual policies into a BR to any distribution of said policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training.

The generalized Mixed-Opponent algorithm is shown in Algorithm 3. We employ Q-Mixing (Equation 1) as our Combine-Oponents.

An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players.

We run both algorithms on the Gathering-Open game with three players.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

Next, we introduce Mixed-Opponents which modifies the BR training by constructing a single opponent policy that represents an aggregate of the mixed strategy and learns a response to this new opponent. Mixed-Opponents also trivially scales to games with more than two players.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Let me evaluate the claim that "Mixed-Opponents applies directly over the joint opponent space, and so has the potential to scale beyond two-player games."

Analysis:
The evidence supports this claim in several ways:
1. The text explicitly states "Mixed-Opponents...trivially scales to games with more than two players"
2. The authors demonstrate this by running experiments on a three-player game (Gathering-Open)
3. The algorithm description shows it uses Q-Mixing to combine opponent strategies into a single policy, which can handle multiple opponents
4. The results show successful performance in both two and three-player scenarios

The claim is well-stated as it:
- Uses measured language ("potential to scale")
- Is directly supported by both theoretical description and experimental evidence
- Makes no exaggerated claims about performance or capabilities

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

Mixed-Oppennts applies directly over the joint opponent space, and so has the potential to scale beyond two-player games.

The review comment to be evaluated is:

The paper suggests two techniques to improve the calculation of empirically figuring out a Nash equilibrium using an iterative application of best-response dynamics. One method learns the best-response to the previously used strategy. The other uses that technique to model the opponent, and then best-responds to the modeled opponent. The experiments show a faster reaching to NE than without these changes.

The paper is well-written and explained, and is accessible even to researchers not well-versed in ML topics. While, the suggested changes are rather straightforward, they do indeed lead to the expected advance (shorter times to reach NE). I was convinced by further introspection that this is a significant enough contribution, to merit acceptance.

Of course, a more significant change to the algorithm, leading not only to a shorter time but to convergence to better equilibria (in cases where multiple exist) would be far more compelling.

The evidence to be evaluated is:A variety of other MSSs have been proposed, and assessing their relative merits is a topic of active research (Balduzzi et al., 2019). In experiments reported here, we adopt NE as the baseline MSS; however, our methods readily apply to any MSS.

In Section 3.2 we present a second algorithm that avoids responding to similar opponent strategies on subsequent iterations, while also addressing the opponent variance issue and providing scalability to multiple other agents.

During each epoch of Mixed-Oracles, a BR is learned for the single new opponent policy, rather than for the mixed opponent-profile generated by the MSS. A BR to the MSS-generated target mixture is then _constructed_ from the collection of BR results for constituent policies in the mixture. Constructing the new policy is done through a general Combine-Respones function that maps a set of policies and a distribution over the policies into a single policy. The resulting policy should approximately aggregate the behavior of the policies.

The key design choice is how to combine knowledge from the BRs to individual policies into a BR to any distribution of said policies.

Smith et al. (2020) propose _Q-Mixing_ as an approach for constructing policies against any mixture of opponent strategies. The method employs Q-values learned against each individual opponent strategy, and thus can support the transfer sought here. Specifically, Q-Mixing averages the Q-values learned against each opponent policy \(\pi_{-i}\), weighted by their likelihood in the opponent mixture \(\sigma_{-i}\): \[Q_{i}(o_{i},\!a_{i}|\,\sigma_{-i})\!=\!\sum_{\pi_{-i}}\!\psi_{i}(\pi_{-i}|\,o_ {i},\!\sigma_{-i})Q_{i}(o_{i},\!a_{i}|\,\pi_{-i}), \tag{1}\] where \(\psi\) determines the relative likelihood of playing an opponent \(\psi_{i}\colon\mathcal{O}_{i}\rightarrow\Delta(\Pi_{-i})\). In this study, we use Q-Mixing as our Combine-Respones, where \(\psi\) is the prior over the opponent distribution as given by an MSS.

This method also employs a combination method, but instead of combining results from training against previously encountered opponents, it combines the strategies of the opponent mixture themselves to construct a single new opponent policy as a target for training.

The generalized Mixed-Opponent algorithm is shown in Algorithm 3. We employ Q-Mixing (Equation 1) as our Combine-Oponents.

An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players.

We run both algorithms on the Gathering-Open game with three players.

Mixed-Oracles and Mixed-Opponents find solutions at least as good as PSRO in fewer timesteps, confirming our hypothesis. In particular, under the pure-hparams, where simulations are more limited, Mixed-Opponents find a better solution than PSRO in half the simulation timesteps.

Next, we introduce Mixed-Opponents which modifies the BR training by constructing a single opponent policy that represents an aggregate of the mixed strategy and learns a response to this new opponent. Mixed-Opponents also trivially scales to games with more than two players.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll evaluate the claim "Mixed-Opponents applies directly over the joint opponent space, and so has the potential to scale beyond two-player games."

Analysis:
The evidence provides several supporting points:
1. The paper explicitly states "An advantage of Mixed-Opponents over Mixed-Oracles is that it can be applied to games with more than two players."
2. The authors demonstrate this by running experiments on a three-player game ("We run both algorithms on the Gathering-Open game with three players")
3. The method uses Q-Mixing as Combine-Opponents which can handle opponent mixtures, supporting scalability to multiple opponents
4. They show practical results with the three-player game where "Mixed-Opponents find a better solution than PSRO in half the simulation timesteps"

The claim is well-stated (score 1) as it:
- Uses measured language ("has the potential to scale")
- Is directly supported by both theoretical explanation and experimental evidence
- Demonstrates actual implementation beyond two players
- Shows practical benefits in multi-player scenarios

<score>1</score>